src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
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(*  Title:      HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
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    Author:     Amine Chaieb
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*)
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header{* A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008 *}
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theory Parametric_Ferrante_Rackoff
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imports Reflected_Multivariate_Polynomial Dense_Linear_Order DP_Library
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  "~~/src/HOL/Library/Efficient_Nat" "~~/src/HOL/Library/Old_Recdef"
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begin
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subsection {* Terms *}
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datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm 
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  | Neg tm | Sub tm tm | CNP nat poly tm
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  (* A size for poly to make inductive proofs simpler*)
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primrec tmsize :: "tm \<Rightarrow> nat" where
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  "tmsize (CP c) = polysize c"
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| "tmsize (Bound n) = 1"
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| "tmsize (Neg a) = 1 + tmsize a"
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| "tmsize (Add a b) = 1 + tmsize a + tmsize b"
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| "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
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| "tmsize (Mul c a) = 1 + polysize c + tmsize a"
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| "tmsize (CNP n c a) = 3 + polysize c + tmsize a "
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  (* Semantics of terms tm *)
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primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" where
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  "Itm vs bs (CP c) = (Ipoly vs c)"
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| "Itm vs bs (Bound n) = bs!n"
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| "Itm vs bs (Neg a) = -(Itm vs bs a)"
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| "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
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| "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
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| "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
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| "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"   
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fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
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  "allpolys P (CP c) = P c"
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| "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
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| "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
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| "allpolys P (Neg p) = allpolys P p"
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| "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
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| "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
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| "allpolys P p = True"
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primrec tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool" where
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  "tmboundslt n (CP c) = True"
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| "tmboundslt n (Bound m) = (m < n)"
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| "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
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| "tmboundslt n (Neg a) = tmboundslt n a"
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| "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
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| "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" 
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| "tmboundslt n (Mul i a) = tmboundslt n a"
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primrec tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) where
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  "tmbound0 (CP c) = True"
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| "tmbound0 (Bound n) = (n>0)"
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| "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
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| "tmbound0 (Neg a) = tmbound0 a"
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| "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
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| "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" 
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| "tmbound0 (Mul i a) = tmbound0 a"
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lemma tmbound0_I:
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  assumes nb: "tmbound0 a"
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  shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
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using nb
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by (induct a rule: tm.induct,auto)
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primrec tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) where
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  "tmbound n (CP c) = True"
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| "tmbound n (Bound m) = (n \<noteq> m)"
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| "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
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| "tmbound n (Neg a) = tmbound n a"
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| "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
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| "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" 
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| "tmbound n (Mul i a) = tmbound n a"
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lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)
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lemma tmbound_I: 
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  assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
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  shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
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  using nb le bnd
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  by (induct t rule: tm.induct , auto)
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fun decrtm0:: "tm \<Rightarrow> tm" where
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  "decrtm0 (Bound n) = Bound (n - 1)"
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| "decrtm0 (Neg a) = Neg (decrtm0 a)"
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| "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
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| "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
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| "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
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| "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
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| "decrtm0 a = a"
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fun incrtm0:: "tm \<Rightarrow> tm" where
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  "incrtm0 (Bound n) = Bound (n + 1)"
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| "incrtm0 (Neg a) = Neg (incrtm0 a)"
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| "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
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| "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
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| "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
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| "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
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| "incrtm0 a = a"
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lemma decrtm0: assumes nb: "tmbound0 t"
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  shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
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  using nb by (induct t rule: decrtm0.induct, simp_all)
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lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
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  by (induct t rule: decrtm0.induct, simp_all)
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primrec decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" where
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  "decrtm m (CP c) = (CP c)"
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| "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
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| "decrtm m (Neg a) = Neg (decrtm m a)"
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| "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
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| "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
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| "decrtm m (Mul c a) = Mul c (decrtm m a)"
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| "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"
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primrec removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "removen n [] = []"
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| "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"
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lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
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  by (induct xs arbitrary: n, auto)
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lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
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  by (induct xs arbitrary: n, auto)
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lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
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  by (induct xs arbitrary: n, auto)
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lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m 
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  else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
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proof(induct xs arbitrary: n m)
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  case Nil thus ?case by simp
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next
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  case (Cons x xs n m)
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  {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
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  moreover
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  {assume nxs: "\<not> (n \<ge> length (x#xs))" 
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    {assume mln: "m < n" hence ?case using Cons by (cases m, auto)}
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    moreover
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    {assume mln: "\<not> (m < n)" 
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      {assume mxs: "m \<le> length (x#xs)" hence ?case using Cons by (cases m, auto)}
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      moreover
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      {assume mxs: "\<not> (m \<le> length (x#xs))" 
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        have th: "length (removen n (x#xs)) = length xs" 
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          using removen_length[where n="n" and xs="x#xs"] nxs by simp
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        with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
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        hence "(removen n (x#xs))!m = [] ! (m - length xs)" 
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          using th nth_length_exceeds[OF mxs'] by auto
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        hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" 
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          by auto
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        hence ?case using nxs mln mxs by auto }
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      ultimately have ?case by blast
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    }
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    ultimately have ?case by blast
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  } ultimately show ?case by blast
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qed
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lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" 
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  and nle: "m \<le> length bs" 
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  shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
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  using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth)
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primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" where
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  "tmsubst0 t (CP c) = CP c"
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| "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
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| "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
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| "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
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| "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
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| "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" 
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| "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
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lemma tmsubst0:
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  shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
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  by (induct a rule: tm.induct) auto
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lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
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  by (induct a rule: tm.induct) auto
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primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" where
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  "tmsubst n t (CP c) = CP c"
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| "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
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| "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a) 
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             else CNP m c (tmsubst n t a))"
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| "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
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| "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
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| "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" 
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| "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"
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lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
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  shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
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using nb nlt
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by (induct a rule: tm.induct,auto)
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lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
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shows "tmbound0 (tmsubst 0 t a)"
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using tnb
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by (induct a rule: tm.induct, auto)
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lemma tmsubst_nb: assumes tnb: "tmbound m t"
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shows "tmbound m (tmsubst m t a)"
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using tnb
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by (induct a rule: tm.induct, auto)
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lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
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  by (induct t, auto)
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  (* Simplification *)
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consts
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  tmadd:: "tm \<times> tm \<Rightarrow> tm"
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recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
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  "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
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  (if n1=n2 then 
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  (let c = c1 +\<^sub>p c2
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  in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
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  else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) 
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  else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
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  "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"  
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  "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))" 
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  "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
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  "tmadd (a,b) = Add a b"
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lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
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apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
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apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
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apply (case_tac "n1 = n2", simp_all add: field_simps)
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apply (simp only: right_distrib[symmetric]) 
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by (auto simp del: polyadd simp add: polyadd[symmetric])
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lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
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by (induct t s rule: tmadd.induct, auto simp add: Let_def)
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lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
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by (induct t s rule: tmadd.induct, auto simp add: Let_def)
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lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
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by (induct t s rule: tmadd.induct, auto simp add: Let_def)
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lemma tmadd_allpolys_npoly[simp]: "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmadd(t,s))" by (induct t s rule: tmadd.induct, simp_all add: Let_def polyadd_norm)
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fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm" where
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  "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
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| "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
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| "tmmul t = (\<lambda> i. Mul i t)"
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lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
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by (induct t arbitrary: i rule: tmmul.induct, simp_all add: field_simps)
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lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
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by (induct t arbitrary: i rule: tmmul.induct, auto )
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lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
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by (induct t arbitrary: n rule: tmmul.induct, auto )
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lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
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by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)
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lemma tmmul_allpolys_npoly[simp]: 
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  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "allpolys isnpoly t \<Longrightarrow> isnpoly c \<Longrightarrow> allpolys isnpoly (tmmul t c)" by (induct t rule: tmmul.induct, simp_all add: Let_def polymul_norm)
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definition tmneg :: "tm \<Rightarrow> tm" where
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  "tmneg t \<equiv> tmmul t (C (- 1,1))"
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definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" where
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  "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"
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lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
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using tmneg_def[of t] 
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apply simp
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done
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lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
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using tmneg_def by simp
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lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
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using tmneg_def by simp
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lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
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using tmneg_def by simp
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lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
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lemma tmneg_allpolys_npoly[simp]: 
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  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" 
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  unfolding tmneg_def by auto
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lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
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using tmsub_def by simp
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lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
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using tmsub_def by simp
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lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
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using tmsub_def by simp
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lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
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using tmsub_def by simp
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lemma tmsub_allpolys_npoly[simp]: 
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  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" 
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  unfolding tmsub_def by (simp add: isnpoly_def)
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fun simptm:: "tm \<Rightarrow> tm" where
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  "simptm (CP j) = CP (polynate j)"
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| "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
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| "simptm (Neg t) = tmneg (simptm t)"
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| "simptm (Add t s) = tmadd (simptm t,simptm s)"
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| "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
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| "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
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| "simptm (CNP n c t) = (let c' = polynate c in if c' = 0\<^sub>p then simptm t else tmadd (CNP n c' (CP 0\<^sub>p ), simptm t))"
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lemma polynate_stupid: 
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  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a)"
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apply (subst polynate[symmetric])
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apply simp
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done
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lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
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by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid) 
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lemma simptm_tmbound0[simp]: 
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  "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
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by (induct t rule: simptm.induct, auto simp add: Let_def)
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lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
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by (induct t rule: simptm.induct, auto simp add: Let_def)
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lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
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by (induct t rule: simptm.induct, auto simp add: Let_def)
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lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))" 
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  by (simp_all add: isnpoly_def)
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lemma simptm_allpolys_npoly[simp]: 
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  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "allpolys isnpoly (simptm p)"
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  by (induct p rule: simptm.induct, auto simp add: Let_def)
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declare let_cong[fundef_cong del]
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fun split0 :: "tm \<Rightarrow> (poly \<times> tm)" where
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  "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
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| "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
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| "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
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| "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
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| "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
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| "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
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| "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
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| "split0 t = (0\<^sub>p, t)"
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declare let_cong[fundef_cong]
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lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
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  apply (rule exI[where x="fst (split0 p)"])
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  apply (rule exI[where x="snd (split0 p)"])
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  by simp
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lemma split0:
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  "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
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  apply (induct t rule: split0.induct)
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  apply simp
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  apply (simp add: Let_def split_def field_simps)
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  apply (simp add: Let_def split_def field_simps)
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  apply (simp add: Let_def split_def field_simps)
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  apply (simp add: Let_def split_def field_simps)
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  apply (simp add: Let_def split_def field_simps)
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  apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
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  apply (simp add: Let_def split_def field_simps)
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  apply (simp add: Let_def split_def field_simps)
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  done
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lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
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proof-
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  fix c' t'
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  assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
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  with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
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qed
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lemma split0_nb0: 
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  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
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proof-
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  fix c' t'
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  assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
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  with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
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qed
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lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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  shows "tmbound0 (snd (split0 t))"
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   384
  using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
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diff changeset
   386
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   387
lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   388
  using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   389
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   390
lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   391
  using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   392
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   393
lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   394
 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   395
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   396
lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   397
by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   398
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   399
lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   400
 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   401
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   402
lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   403
by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   404
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   405
lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   406
  shows 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   407
  "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   408
  by (induct p rule: split0.induct, 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   409
    auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   410
    Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   411
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   412
subsection{* Formulae *}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   413
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   414
datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   415
  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   416
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   417
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   418
  (* A size for fm *)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   419
fun fmsize :: "fm \<Rightarrow> nat" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   420
  "fmsize (NOT p) = 1 + fmsize p"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   421
| "fmsize (And p q) = 1 + fmsize p + fmsize q"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   422
| "fmsize (Or p q) = 1 + fmsize p + fmsize q"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   423
| "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   424
| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   425
| "fmsize (E p) = 1 + fmsize p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   426
| "fmsize (A p) = 4+ fmsize p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   427
| "fmsize p = 1"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   428
  (* several lemmas about fmsize *)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   429
lemma fmsize_pos[termination_simp]: "fmsize p > 0"        
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   430
by (induct p rule: fmsize.induct) simp_all
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   431
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   432
  (* Semantics of formulae (fm) *)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 45499
diff changeset
   433
primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   434
  "Ifm vs bs T = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   435
| "Ifm vs bs F = False"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   436
| "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   437
| "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   438
| "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   439
| "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   440
| "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   441
| "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   442
| "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   443
| "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   444
| "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   445
| "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   446
| "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   447
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   448
fun not:: "fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   449
  "not (NOT (NOT p)) = not p"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   450
| "not (NOT p) = p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   451
| "not T = F"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   452
| "not F = T"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   453
| "not (Lt t) = Le (tmneg t)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   454
| "not (Le t) = Lt (tmneg t)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   455
| "not (Eq t) = NEq t"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   456
| "not (NEq t) = Eq t"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   457
| "not p = NOT p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   458
lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   459
by (induct p rule: not.induct) auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   460
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   461
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   462
  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   463
   if p = q then p else And p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   464
lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   465
by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   466
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   467
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   468
  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   469
       else if p=q then p else Or p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   470
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   471
lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   472
by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   473
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   474
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   475
  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   476
    else Imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   477
lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   478
by (cases "p=F \<or> q=T",simp_all add: imp_def) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   479
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   480
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   481
  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   482
       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 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   483
  Iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   484
lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   485
  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   486
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   487
  (* Quantifier freeness *)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   488
fun qfree:: "fm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   489
  "qfree (E p) = False"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   490
| "qfree (A p) = False"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   491
| "qfree (NOT p) = qfree p" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   492
| "qfree (And p q) = (qfree p \<and> qfree q)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   493
| "qfree (Or  p q) = (qfree p \<and> qfree q)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   494
| "qfree (Imp p q) = (qfree p \<and> qfree q)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   495
| "qfree (Iff p q) = (qfree p \<and> qfree q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   496
| "qfree p = True"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   497
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   498
  (* Boundedness and substitution *)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   499
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   500
primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   501
  "boundslt n T = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   502
| "boundslt n F = True"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   503
| "boundslt n (Lt t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   504
| "boundslt n (Le t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   505
| "boundslt n (Eq t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   506
| "boundslt n (NEq t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   507
| "boundslt n (NOT p) = boundslt n p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   508
| "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   509
| "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   510
| "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   511
| "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   512
| "boundslt n (E p) = boundslt (Suc n) p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   513
| "boundslt n (A p) = boundslt (Suc n) p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   514
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   515
fun bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   516
  "bound0 T = True"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   517
| "bound0 F = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   518
| "bound0 (Lt a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   519
| "bound0 (Le a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   520
| "bound0 (Eq a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   521
| "bound0 (NEq a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   522
| "bound0 (NOT p) = bound0 p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   523
| "bound0 (And p q) = (bound0 p \<and> bound0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   524
| "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   525
| "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   526
| "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   527
| "bound0 p = False"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   528
lemma bound0_I:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   529
  assumes bp: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   530
  shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   531
using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   532
by (induct p rule: bound0.induct,auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   533
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   534
primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   535
  "bound m T = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   536
| "bound m F = True"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   537
| "bound m (Lt t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   538
| "bound m (Le t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   539
| "bound m (Eq t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   540
| "bound m (NEq t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   541
| "bound m (NOT p) = bound m p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   542
| "bound m (And p q) = (bound m p \<and> bound m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   543
| "bound m (Or p q) = (bound m p \<and> bound m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   544
| "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   545
| "bound m (Iff p q) = (bound m p \<and> bound m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   546
| "bound m (E p) = bound (Suc m) p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   547
| "bound m (A p) = bound (Suc m) p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   548
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   549
lemma bound_I:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   550
  assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   551
  shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   552
  using bnd nb le tmbound_I[where bs=bs and vs = vs]
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   553
proof(induct p arbitrary: bs n rule: fm.induct)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   554
  case (E p bs n) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   555
  {fix y
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   556
    from E have bnd: "boundslt (length (y#bs)) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   557
      and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   558
    from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   559
  thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   560
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   561
  case (A p bs n) {fix y
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   562
    from A have bnd: "boundslt (length (y#bs)) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   563
      and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   564
    from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   565
  thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   566
qed auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   567
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   568
fun decr0 :: "fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   569
  "decr0 (Lt a) = Lt (decrtm0 a)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   570
| "decr0 (Le a) = Le (decrtm0 a)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   571
| "decr0 (Eq a) = Eq (decrtm0 a)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   572
| "decr0 (NEq a) = NEq (decrtm0 a)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   573
| "decr0 (NOT p) = NOT (decr0 p)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   574
| "decr0 (And p q) = conj (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   575
| "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   576
| "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   577
| "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   578
| "decr0 p = p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   579
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   580
lemma decr0: assumes nb: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   581
  shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   582
  using nb 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   583
  by (induct p rule: decr0.induct, simp_all add: decrtm0)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   584
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   585
primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   586
  "decr m T = T"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   587
| "decr m F = F"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   588
| "decr m (Lt t) = (Lt (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   589
| "decr m (Le t) = (Le (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   590
| "decr m (Eq t) = (Eq (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   591
| "decr m (NEq t) = (NEq (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   592
| "decr m (NOT p) = NOT (decr m p)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   593
| "decr m (And p q) = conj (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   594
| "decr m (Or p q) = disj (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   595
| "decr m (Imp p q) = imp (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   596
| "decr m (Iff p q) = iff (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   597
| "decr m (E p) = E (decr (Suc m) p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   598
| "decr m (A p) = A (decr (Suc m) p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   599
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   600
lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   601
  and nle: "m < length bs" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   602
  shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   603
  using bnd nb nle
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   604
proof(induct p arbitrary: bs m rule: fm.induct)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   605
  case (E p bs m) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   606
  {fix x
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   607
    from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   608
  and nle: "Suc m < length (x#bs)" by auto
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   609
    from E(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   610
  } thus ?case by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   611
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   612
  case (A p bs m)  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   613
  {fix x
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   614
    from A have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   615
  and nle: "Suc m < length (x#bs)" by auto
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   616
    from A(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   617
  } thus ?case by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   618
qed (auto simp add: decrtm removen_nth)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   619
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   620
primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   621
  "subst0 t T = T"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   622
| "subst0 t F = F"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   623
| "subst0 t (Lt a) = Lt (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   624
| "subst0 t (Le a) = Le (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   625
| "subst0 t (Eq a) = Eq (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   626
| "subst0 t (NEq a) = NEq (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   627
| "subst0 t (NOT p) = NOT (subst0 t p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   628
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   629
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   630
| "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   631
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   632
| "subst0 t (E p) = E p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   633
| "subst0 t (A p) = A p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   634
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   635
lemma subst0: assumes qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   636
  shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   637
using qf tmsubst0[where x="x" and bs="bs" and t="t"]
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   638
by (induct p rule: fm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   639
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   640
lemma subst0_nb:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   641
  assumes bp: "tmbound0 t" and qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   642
  shows "bound0 (subst0 t p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   643
using qf tmsubst0_nb[OF bp] bp
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   644
by (induct p rule: fm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   645
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   646
primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   647
  "subst n t T = T"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   648
| "subst n t F = F"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   649
| "subst n t (Lt a) = Lt (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   650
| "subst n t (Le a) = Le (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   651
| "subst n t (Eq a) = Eq (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   652
| "subst n t (NEq a) = NEq (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   653
| "subst n t (NOT p) = NOT (subst n t p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   654
| "subst n t (And p q) = And (subst n t p) (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   655
| "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   656
| "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   657
| "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   658
| "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   659
| "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   660
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   661
lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   662
  shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   663
  using nb nlm
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   664
proof (induct p arbitrary: bs n t rule: fm.induct)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   665
  case (E p bs n) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   666
  {fix x 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   667
    from E have bn: "boundslt (length (x#bs)) p" by simp 
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   668
    from E have nlm: "Suc n \<le> length (x#bs)" by simp
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   669
    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 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   670
    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   671
    by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   672
thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   673
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   674
  case (A p bs n)   
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   675
  {fix x 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   676
    from A have bn: "boundslt (length (x#bs)) p" by simp 
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   677
    from A have nlm: "Suc n \<le> length (x#bs)" by simp
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   678
    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 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   679
    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   680
    by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   681
thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   682
qed(auto simp add: tmsubst)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   683
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   684
lemma subst_nb: assumes tnb: "tmbound m t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   685
shows "bound m (subst m t p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   686
using tnb tmsubst_nb incrtm0_tmbound
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   687
by (induct p arbitrary: m t rule: fm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   688
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   689
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   690
by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   691
lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   692
by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   693
lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   694
by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   695
lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   696
 by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   697
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   698
lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   699
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   700
lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   701
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   702
lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   703
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   704
lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   705
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   706
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   707
lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   708
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   709
lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   710
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   711
lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   712
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   713
lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   714
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   715
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   716
lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   717
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   718
lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   719
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   720
lemma imp_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   721
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   722
lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   723
using imp_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   724
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   725
lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   726
  by (unfold iff_def,cases "p=q", auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   727
lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   728
using iff_def by (unfold iff_def,cases "p=q", auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   729
lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   730
using iff_def by (unfold iff_def,cases "p=q", auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   731
lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   732
using iff_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   733
lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   734
by (induct p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   735
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   736
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   737
  "isatom T = True"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   738
| "isatom F = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   739
| "isatom (Lt a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   740
| "isatom (Le a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   741
| "isatom (Eq a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   742
| "isatom (NEq a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   743
| "isatom p = False"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   744
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   745
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   746
by (induct p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   747
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   748
definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   749
  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   750
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   751
definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   752
  "evaldjf f ps \<equiv> foldr (djf f) ps F"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   753
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   754
lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   755
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   756
(cases "f p", simp_all add: Let_def djf_def) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   757
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   758
lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   759
  by(induct ps, simp_all add: evaldjf_def djf_Or)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   760
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   761
lemma evaldjf_bound0: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   762
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   763
  shows "bound0 (evaldjf f xs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   764
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   765
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   766
lemma evaldjf_qf: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   767
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   768
  shows "qfree (evaldjf f xs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   769
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   770
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   771
fun disjuncts :: "fm \<Rightarrow> fm list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   772
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   773
| "disjuncts F = []"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   774
| "disjuncts p = [p]"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   775
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   776
lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   777
by(induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   778
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   779
lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   780
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   781
  assume nb: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   782
  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   783
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   784
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   785
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   786
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   787
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   788
  assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   789
  hence "list_all qfree (disjuncts p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   790
    by (induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   791
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   792
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   793
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   794
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   795
  "DJ f p \<equiv> evaldjf f (disjuncts p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   796
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   797
lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   798
  and fF: "f F = F"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   799
  shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   800
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   801
  have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   802
    by (simp add: DJ_def evaldjf_ex) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   803
  also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   804
  finally show ?thesis .
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   805
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   806
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   807
lemma DJ_qf: assumes 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   808
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   809
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   810
proof(clarify)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   811
  fix  p assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   812
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   813
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   814
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   815
  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   816
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   817
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   818
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   819
lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   820
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   821
proof(clarify)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   822
  fix p::fm and bs
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   823
  assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   824
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   825
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   826
  have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   827
    by (simp add: DJ_def evaldjf_ex)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   828
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   829
  also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   830
  finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   831
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   832
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   833
fun conjuncts :: "fm \<Rightarrow> fm list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   834
  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   835
| "conjuncts T = []"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   836
| "conjuncts p = [p]"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   837
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   838
definition list_conj :: "fm list \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   839
  "list_conj ps \<equiv> foldr conj ps T"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   840
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   841
definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   842
  "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   843
                   in conj (decr0 (list_conj yes)) (f (list_conj no)))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   844
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   845
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   846
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   847
  assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   848
  hence "list_all qfree (conjuncts p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   849
    by (induct p rule: conjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   850
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   851
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   852
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   853
lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   854
by(induct p rule: conjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   855
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   856
lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   857
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   858
  assume nb: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   859
  hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   860
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   861
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   862
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   863
fun islin :: "fm \<Rightarrow> bool" where
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   864
  "islin (And p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   865
| "islin (Or p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   866
| "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   867
| "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   868
| "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   869
| "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   870
| "islin (NOT p) = False"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   871
| "islin (Imp p q) = False"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   872
| "islin (Iff p q) = False"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   873
| "islin p = bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   874
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   875
lemma islin_stupid: assumes nb: "tmbound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   876
  shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   877
  using nb by (cases p, auto, case_tac nat, auto)+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   878
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   879
definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   880
definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
38864
4abe644fcea5 formerly unnamed infix equality now named HOL.eq
haftmann
parents: 38795
diff changeset
   881
definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   882
definition "neq p = not (eq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   883
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   884
lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   885
  apply(simp add: lt_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   886
  apply(cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   887
  apply (case_tac poly, simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   888
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   889
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   890
lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   891
  apply(simp add: le_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   892
  apply(cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   893
  apply (case_tac poly, simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   894
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   895
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   896
lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   897
  apply(simp add: eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   898
  apply(cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   899
  apply (case_tac poly, simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   900
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   901
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   902
lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   903
  by(simp add: neq_def eq)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   904
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   905
lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   906
  apply (simp add: lt_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   907
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   908
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   909
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   910
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   911
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   912
lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   913
  apply (simp add: le_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   914
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   915
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   916
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   917
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   918
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   919
lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   920
  apply (simp add: eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   921
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   922
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   923
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   924
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   925
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   926
lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   927
  apply (simp add: neq_def eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   928
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   929
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   930
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   931
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   932
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   933
definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   934
definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   935
definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   936
definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   937
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   938
lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   939
  shows "islin (simplt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   940
  unfolding simplt_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   941
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   942
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])
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   943
  
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   944
lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   945
  shows "islin (simple t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   946
  unfolding simple_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   947
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   948
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)
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   949
lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   950
  shows "islin (simpeq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   951
  unfolding simpeq_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   952
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   953
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)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   954
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   955
lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   956
  shows "islin (simpneq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   957
  unfolding simpneq_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   958
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   959
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)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   960
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   961
lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   962
  by (cases "split0 s", auto)
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   963
lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   964
  and n: "allpolys isnpoly t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   965
  shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   966
  using n
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   967
  by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   968
lemma simplt[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   969
  shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   970
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   971
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   972
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   973
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   974
      using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   975
      by (simp add: simplt_def Let_def split_def lt)}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   976
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   977
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   978
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   979
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   980
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   981
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   982
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   983
lemma simple[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   984
  shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   985
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   986
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   987
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   988
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   989
      using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   990
      by (simp add: simple_def Let_def split_def le)}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   991
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   992
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   993
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   994
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   995
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   996
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   997
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   998
lemma simpeq[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   999
  shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1000
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1001
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1002
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1003
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1004
      using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1005
      by (simp add: simpeq_def Let_def split_def)}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1006
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1007
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1008
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1009
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1010
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1011
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1012
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1013
lemma simpneq[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1014
  shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1015
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1016
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1017
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1018
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1019
      using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1020
      by (simp add: simpneq_def Let_def split_def )}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1021
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1022
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1023
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1024
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1025
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1026
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1027
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1028
lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1029
  apply (simp add: lt_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1030
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1031
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1032
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1033
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1034
lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1035
  apply (simp add: le_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1036
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1037
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1038
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1039
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1040
lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1041
  apply (simp add: eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1042
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1043
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1044
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1045
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1046
lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1047
  apply (simp add: neq_def eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1048
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1049
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1050
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1051
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1052
lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1053
  shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
48562
f6d6d58fa318 tuned proofs -- avoid odd situations of polymorphic Frees in goal state;
wenzelm
parents: 47108
diff changeset
  1054
  using split0 [of "simptm t" "vs::'a list" bs]
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1055
proof(simp add: simplt_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1056
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1057
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1058
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1059
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1060
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1061
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1062
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1063
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1064
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1065
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1066
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1067
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1068
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1069
lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1070
  shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
48562
f6d6d58fa318 tuned proofs -- avoid odd situations of polymorphic Frees in goal state;
wenzelm
parents: 47108
diff changeset
  1071
  using split0 [of "simptm t" "vs::'a list" bs]
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1072
proof(simp add: simple_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1073
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1074
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1075
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1076
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1077
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1078
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1079
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1080
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1081
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1082
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1083
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1084
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1085
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1086
lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1087
  shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
48562
f6d6d58fa318 tuned proofs -- avoid odd situations of polymorphic Frees in goal state;
wenzelm
parents: 47108
diff changeset
  1088
  using split0 [of "simptm t" "vs::'a list" bs]
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1089
proof(simp add: simpeq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1090
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1091
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1092
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1093
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1094
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1095
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1096
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1097
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1098
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1099
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1100
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1101
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1102
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1103
lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1104
  shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
48562
f6d6d58fa318 tuned proofs -- avoid odd situations of polymorphic Frees in goal state;
wenzelm
parents: 47108
diff changeset
  1105
  using split0 [of "simptm t" "vs::'a list" bs]
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1106
proof(simp add: simpneq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1107
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1108
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1109
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1110
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1111
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1112
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1113
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1114
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1115
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1116
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1117
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1118
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1119
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
  1120
fun conjs   :: "fm \<Rightarrow> fm list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1121
  "conjs (And p q) = (conjs p)@(conjs q)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
  1122
| "conjs T = []"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
  1123
| "conjs p = [p]"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1124
lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1125
by (induct p rule: conjs.induct, auto)
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
  1126
definition list_disj :: "fm list \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1127
  "list_disj ps \<equiv> foldr disj ps F"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1128
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1129
lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1130
  by (induct ps, auto simp add: list_conj_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1131
lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1132
  by (induct ps, auto simp add: list_conj_def conj_qf)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1133
lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1134
  by (induct ps, auto simp add: list_disj_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1135
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1136
lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1137
  unfolding conj_def by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1138
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1139
lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1140
  apply (induct p rule: conjs.induct) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1141
  apply (unfold conjs.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1142
  apply (unfold set_append)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1143
  apply (unfold ball_Un)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1144
  apply (unfold bound.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1145
  apply auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1146
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1147
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1148
lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1149
  apply (induct p rule: conjs.induct) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1150
  apply (unfold conjs.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1151
  apply (unfold set_append)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1152
  apply (unfold ball_Un)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1153
  apply (unfold boundslt.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1154
  apply blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1155
by simp_all
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1156
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1157
lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1158
  unfolding list_conj_def
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1159
  by (induct ps, auto simp add: conj_boundslt)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1160
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1161
lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1162
  shows "bound n (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1163
  using bnd
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1164
  unfolding list_conj_def
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1165
  by (induct ps, auto simp add: conj_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1166
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1167
lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1168
unfolding list_conj_def by (induct ps , auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1169
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1170
lemma CJNB_qe: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1171
  assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1172
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1173
proof(clarify)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1174
  fix bs p
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1175
  assume qfp: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1176
  let ?cjs = "conjuncts p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1177
  let ?yes = "fst (partition bound0 ?cjs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1178
  let ?no = "snd (partition bound0 ?cjs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1179
  let ?cno = "list_conj ?no"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1180
  let ?cyes = "list_conj ?yes"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1181
  have part: "partition bound0 ?cjs = (?yes,?no)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1182
  from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1183
  hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1184
  hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1185
  from conjuncts_qf[OF qfp] partition_set[OF part] 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1186
  have " \<forall>q\<in> set ?no. qfree q" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1187
  hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1188
  with qe have cno_qf:"qfree (qe ?cno )" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1189
    and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1190
  from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1191
    by (simp add: CJNB_def Let_def conj_qf split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1192
  {fix bs