src/HOL/Decision_Procs/MIR.thy
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(*  Title:      HOL/Decision_Procs/MIR.thy
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    Author:     Amine Chaieb
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*)
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theory MIR
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imports Complex_Main Dense_Linear_Order DP_Library
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  "~~/src/HOL/Library/Efficient_Nat"
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uses ("mir_tac.ML")
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begin
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section {* Quantifier elimination for @{text "\<real> (0, 1, +, floor, <)"} *}
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declare real_of_int_floor_cancel [simp del]
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lemma myle: fixes a b :: "'a::{ordered_ab_group_add}"
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  shows "(a \<le> b) = (0 \<le> b - a)"
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by (metis add_0_left add_le_cancel_right diff_add_cancel)
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lemma myless: fixes a b :: "'a::{ordered_ab_group_add}"
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  shows "(a < b) = (0 < b - a)"
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by (metis le_iff_diff_le_0 less_le_not_le myle)
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  (* Maybe should be added to the library \<dots> *)
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lemma floor_int_eq: "(real n\<le> x \<and> x < real (n+1)) = (floor x = n)"
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proof( auto)
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  assume lb: "real n \<le> x"
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    and ub: "x < real n + 1"
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  have "real (floor x) \<le> x" by simp 
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  hence "real (floor x) < real (n + 1) " using ub by arith
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  hence "floor x < n+1" by simp
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  moreover from lb have "n \<le> floor x" using floor_mono[where x="real n" and y="x"] 
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    by simp ultimately show "floor x = n" by simp
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qed
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(* Periodicity of dvd *)
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lemma dvd_period:
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  assumes advdd: "(a::int) dvd d"
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  shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))"
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  using advdd  
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proof-
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  {fix x k
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    from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"]  
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    have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp}
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  hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))"  by simp
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  then show ?thesis by simp
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qed
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(* The Divisibility relation between reals *)
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definition
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  rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50)
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where
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  rdvd_def: "x rdvd y \<longleftrightarrow> (\<exists>k\<Colon>int. y = x * real k)"
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lemma int_rdvd_real: 
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  shows "real (i::int) rdvd x = (i dvd (floor x) \<and> real (floor x) = x)" (is "?l = ?r")
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proof
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  assume "?l" 
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  hence th: "\<exists> k. x=real (i*k)" by (simp add: rdvd_def)
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  hence th': "real (floor x) = x" by (auto simp del: real_of_int_mult)
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  with th have "\<exists> k. real (floor x) = real (i*k)" by simp
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  hence "\<exists> k. floor x = i*k" by (simp only: real_of_int_inject)
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  thus ?r  using th' by (simp add: dvd_def) 
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next
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  assume "?r" hence "(i\<Colon>int) dvd \<lfloor>x\<Colon>real\<rfloor>" ..
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  hence "\<exists> k. real (floor x) = real (i*k)" 
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    by (simp only: real_of_int_inject) (simp add: dvd_def)
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  thus ?l using `?r` by (simp add: rdvd_def)
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qed
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lemma int_rdvd_iff: "(real (i::int) rdvd real t) = (i dvd t)"
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by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only :real_of_int_mult[symmetric])
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lemma rdvd_abs1: 
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  "(abs (real d) rdvd t) = (real (d ::int) rdvd t)"
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proof
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  assume d: "real d rdvd t"
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  from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real (floor t) = t" by auto
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  from iffD2[OF abs_dvd_iff] d2 have "(abs d) dvd (floor t)" by blast
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  with ti int_rdvd_real[symmetric] have "real (abs d) rdvd t" by blast 
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  thus "abs (real d) rdvd t" by simp
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next
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  assume "abs (real d) rdvd t" hence "real (abs d) rdvd t" by simp
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  with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real (floor t) =t" by auto
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  from iffD1[OF abs_dvd_iff] d2 have "d dvd floor t" by blast
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  with ti int_rdvd_real[symmetric] show "real d rdvd t" by blast
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qed
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lemma rdvd_minus: "(real (d::int) rdvd t) = (real d rdvd -t)"
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  apply (auto simp add: rdvd_def)
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  apply (rule_tac x="-k" in exI, simp) 
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  apply (rule_tac x="-k" in exI, simp)
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done
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lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)"
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by (auto simp add: rdvd_def)
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lemma rdvd_mult: 
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  assumes knz: "k\<noteq>0"
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  shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)"
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using knz by (simp add:rdvd_def)
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  (*********************************************************************************)
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  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
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  (*********************************************************************************)
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datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
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  | Mul int num | Floor num| CF int num num
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  (* A size for num to make inductive proofs simpler*)
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primrec num_size :: "num \<Rightarrow> nat" where
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 "num_size (C c) = 1"
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| "num_size (Bound n) = 1"
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| "num_size (Neg a) = 1 + num_size a"
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| "num_size (Add a b) = 1 + num_size a + num_size b"
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| "num_size (Sub a b) = 3 + num_size a + num_size b"
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| "num_size (CN n c a) = 4 + num_size a "
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| "num_size (CF c a b) = 4 + num_size a + num_size b"
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| "num_size (Mul c a) = 1 + num_size a"
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| "num_size (Floor a) = 1 + num_size a"
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  (* Semantics of numeral terms (num) *)
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primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
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  "Inum bs (C c) = (real c)"
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| "Inum bs (Bound n) = bs!n"
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| "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
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| "Inum bs (Neg a) = -(Inum bs a)"
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| "Inum bs (Add a b) = Inum bs a + Inum bs b"
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| "Inum bs (Sub a b) = Inum bs a - Inum bs b"
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| "Inum bs (Mul c a) = (real c) * Inum bs a"
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| "Inum bs (Floor a) = real (floor (Inum bs a))"
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| "Inum bs (CF c a b) = real c * real (floor (Inum bs a)) + Inum bs b"
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definition "isint t bs \<equiv> real (floor (Inum bs t)) = Inum bs t"
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lemma isint_iff: "isint n bs = (real (floor (Inum bs n)) = Inum bs n)"
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by (simp add: isint_def)
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lemma isint_Floor: "isint (Floor n) bs"
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  by (simp add: isint_iff)
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lemma isint_Mul: "isint e bs \<Longrightarrow> isint (Mul c e) bs"
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proof-
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  let ?e = "Inum bs e"
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  let ?fe = "floor ?e"
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  assume be: "isint e bs" hence efe:"real ?fe = ?e" by (simp add: isint_iff)
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  have "real ((floor (Inum bs (Mul c e)))) = real (floor (real (c * ?fe)))" using efe by simp
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  also have "\<dots> = real (c* ?fe)" by (simp only: floor_real_of_int) 
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  also have "\<dots> = real c * ?e" using efe by simp
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  finally show ?thesis using isint_iff by simp
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qed
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lemma isint_neg: "isint e bs \<Longrightarrow> isint (Neg e) bs"
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proof-
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  let ?I = "\<lambda> t. Inum bs t"
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  assume ie: "isint e bs"
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  hence th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)  
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  have "real (floor (?I (Neg e))) = real (floor (- (real (floor (?I e)))))" by (simp add: th)
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  also have "\<dots> = - real (floor (?I e))" by(simp add: floor_minus_real_of_int) 
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  finally show "isint (Neg e) bs" by (simp add: isint_def th)
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qed
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lemma isint_sub: 
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  assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs"
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proof-
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  let ?I = "\<lambda> t. Inum bs t"
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  from ie have th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)  
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  have "real (floor (?I (Sub (C c) e))) = real (floor ((real (c -floor (?I e)))))" by (simp add: th)
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  also have "\<dots> = real (c- floor (?I e))" by(simp add: floor_minus_real_of_int) 
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  finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th)
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qed
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lemma isint_add: assumes
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  ai:"isint a bs" and bi: "isint b bs" shows "isint (Add a b) bs"
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proof-
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  let ?a = "Inum bs a"
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  let ?b = "Inum bs b"
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  from ai bi isint_iff have "real (floor (?a + ?b)) = real (floor (real (floor ?a) + real (floor ?b)))" by simp
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  also have "\<dots> = real (floor ?a) + real (floor ?b)" by simp
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  also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp
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  finally show "isint (Add a b) bs" by (simp add: isint_iff)
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qed
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lemma isint_c: "isint (C j) bs"
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  by (simp add: isint_iff)
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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    (* FORMULAE *)
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datatype fm  = 
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  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
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  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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  (* A size for fm *)
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fun fmsize :: "fm \<Rightarrow> nat" where
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 "fmsize (NOT p) = 1 + fmsize p"
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| "fmsize (And p q) = 1 + fmsize p + fmsize q"
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| "fmsize (Or p q) = 1 + fmsize p + fmsize q"
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| "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
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| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
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| "fmsize (E p) = 1 + fmsize p"
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| "fmsize (A p) = 4+ fmsize p"
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| "fmsize (Dvd i t) = 2"
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| "fmsize (NDvd i t) = 2"
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| "fmsize p = 1"
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  (* several lemmas about fmsize *)
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lemma fmsize_pos: "fmsize p > 0"
23264
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by (induct p rule: fmsize.induct) simp_all
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  (* Semantics of formulae (fm) *)
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primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
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  "Ifm bs T = True"
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| "Ifm bs F = False"
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| "Ifm bs (Lt a) = (Inum bs a < 0)"
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| "Ifm bs (Gt a) = (Inum bs a > 0)"
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| "Ifm bs (Le a) = (Inum bs a \<le> 0)"
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| "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
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| "Ifm bs (Eq a) = (Inum bs a = 0)"
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| "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
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| "Ifm bs (Dvd i b) = (real i rdvd Inum bs b)"
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| "Ifm bs (NDvd i b) = (\<not>(real i rdvd Inum bs b))"
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| "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
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| "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
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| "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
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| "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
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| "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
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| "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
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| "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
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   229
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consts prep :: "fm \<Rightarrow> fm"
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recdef prep "measure fmsize"
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  "prep (E T) = T"
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  "prep (E F) = F"
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chaieb
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  "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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  "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
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   236
  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
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   237
  "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   238
  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
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chaieb
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   239
  "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   240
  "prep (E p) = E (prep p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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  "prep (A (And p q)) = And (prep (A p)) (prep (A q))"
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   242
  "prep (A p) = prep (NOT (E (NOT p)))"
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   243
  "prep (NOT (NOT p)) = prep p"
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   244
  "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))"
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   245
  "prep (NOT (A p)) = prep (E (NOT p))"
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   246
  "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))"
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   247
  "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))"
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   248
  "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))"
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   249
  "prep (NOT p) = NOT (prep p)"
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   250
  "prep (Or p q) = Or (prep p) (prep q)"
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   251
  "prep (And p q) = And (prep p) (prep q)"
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   252
  "prep (Imp p q) = prep (Or (NOT p) q)"
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   253
  "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))"
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   254
  "prep p = p"
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   255
(hints simp add: fmsize_pos)
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   256
lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
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   257
by (induct p rule: prep.induct, auto)
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   258
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   259
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   260
  (* Quantifier freeness *)
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   261
fun qfree:: "fm \<Rightarrow> bool" where
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   262
  "qfree (E p) = False"
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   263
  | "qfree (A p) = False"
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   264
  | "qfree (NOT p) = qfree p" 
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   265
  | "qfree (And p q) = (qfree p \<and> qfree q)" 
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   266
  | "qfree (Or  p q) = (qfree p \<and> qfree q)" 
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   267
  | "qfree (Imp p q) = (qfree p \<and> qfree q)" 
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   268
  | "qfree (Iff p q) = (qfree p \<and> qfree q)"
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   269
  | "qfree p = True"
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   270
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   271
  (* Boundedness and substitution *)
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primrec numbound0 :: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
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   273
  "numbound0 (C c) = True"
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   274
  | "numbound0 (Bound n) = (n>0)"
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   275
  | "numbound0 (CN n i a) = (n > 0 \<and> numbound0 a)"
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   276
  | "numbound0 (Neg a) = numbound0 a"
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   277
  | "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
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   278
  | "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
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   279
  | "numbound0 (Mul i a) = numbound0 a"
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   280
  | "numbound0 (Floor a) = numbound0 a"
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   281
  | "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)" 
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   282
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   283
lemma numbound0_I:
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  assumes nb: "numbound0 a"
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   285
  shows "Inum (b#bs) a = Inum (b'#bs) a"
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   286
  using nb by (induct a) auto
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   287
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   288
lemma numbound0_gen: 
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   289
  assumes nb: "numbound0 t" and ti: "isint t (x#bs)"
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   290
  shows "\<forall> y. isint t (y#bs)"
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chaieb
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   291
using nb ti 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   292
proof(clarify)
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   293
  fix y
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   294
  from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def]
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   295
  show "isint t (y#bs)"
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   296
    by (simp add: isint_def)
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chaieb
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   297
qed
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   298
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   299
primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
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   300
  "bound0 T = True"
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   301
  | "bound0 F = True"
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   302
  | "bound0 (Lt a) = numbound0 a"
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   303
  | "bound0 (Le a) = numbound0 a"
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   304
  | "bound0 (Gt a) = numbound0 a"
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   305
  | "bound0 (Ge a) = numbound0 a"
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   306
  | "bound0 (Eq a) = numbound0 a"
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   307
  | "bound0 (NEq a) = numbound0 a"
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   308
  | "bound0 (Dvd i a) = numbound0 a"
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   309
  | "bound0 (NDvd i a) = numbound0 a"
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   310
  | "bound0 (NOT p) = bound0 p"
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   311
  | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
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   312
  | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
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   313
  | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
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   314
  | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
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   315
  | "bound0 (E p) = False"
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   316
  | "bound0 (A p) = False"
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   317
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   318
lemma bound0_I:
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   319
  assumes bp: "bound0 p"
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   320
  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
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   321
 using bp numbound0_I [where b="b" and bs="bs" and b'="b'"]
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   322
  by (induct p) auto
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   323
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   324
primrec numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) where
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   325
  "numsubst0 t (C c) = (C c)"
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   326
  | "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
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   327
  | "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))"
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   328
  | "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)"
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haftmann
parents: 25162
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   329
  | "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
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   330
  | "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
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   331
  | "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" 
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   332
  | "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"
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   333
  | "numsubst0 t (Floor a) = Floor (numsubst0 t a)"
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diff changeset
   334
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   335
lemma numsubst0_I:
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   336
  shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
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   337
  by (induct t) simp_all
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diff changeset
   338
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   339
primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) where
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   340
  "subst0 t T = T"
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   341
  | "subst0 t F = F"
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   342
  | "subst0 t (Lt a) = Lt (numsubst0 t a)"
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   343
  | "subst0 t (Le a) = Le (numsubst0 t a)"
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diff changeset
   344
  | "subst0 t (Gt a) = Gt (numsubst0 t a)"
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   345
  | "subst0 t (Ge a) = Ge (numsubst0 t a)"
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diff changeset
   346
  | "subst0 t (Eq a) = Eq (numsubst0 t a)"
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diff changeset
   347
  | "subst0 t (NEq a) = NEq (numsubst0 t a)"
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   348
  | "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   349
  | "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   350
  | "subst0 t (NOT p) = NOT (subst0 t p)"
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haftmann
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diff changeset
   351
  | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
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haftmann
parents: 25162
diff changeset
   352
  | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
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haftmann
parents: 25162
diff changeset
   353
  | "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)"
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diff changeset
   354
  | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
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   355
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   356
lemma subst0_I: assumes qfp: "qfree p"
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diff changeset
   357
  shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p"
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chaieb
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diff changeset
   358
  using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
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diff changeset
   359
  by (induct p) simp_all
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diff changeset
   360
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   361
fun decrnum:: "num \<Rightarrow> num" where
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chaieb
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   362
  "decrnum (Bound n) = Bound (n - 1)"
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   363
| "decrnum (Neg a) = Neg (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
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diff changeset
   364
| "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   365
| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
421a795cee05 recdef -> fun(ction)
krauss
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diff changeset
   366
| "decrnum (Mul c a) = Mul c (decrnum a)"
421a795cee05 recdef -> fun(ction)
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   367
| "decrnum (Floor a) = Floor (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
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diff changeset
   368
| "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   369
| "decrnum (CF c a b) = CF c (decrnum a) (decrnum b)"
421a795cee05 recdef -> fun(ction)
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   370
| "decrnum a = a"
421a795cee05 recdef -> fun(ction)
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diff changeset
   371
421a795cee05 recdef -> fun(ction)
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   372
fun decr :: "fm \<Rightarrow> fm" where
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parents:
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   373
  "decr (Lt a) = Lt (decrnum a)"
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krauss
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   374
| "decr (Le a) = Le (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
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diff changeset
   375
| "decr (Gt a) = Gt (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   376
| "decr (Ge a) = Ge (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   377
| "decr (Eq a) = Eq (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   378
| "decr (NEq a) = NEq (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   379
| "decr (Dvd i a) = Dvd i (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   380
| "decr (NDvd i a) = NDvd i (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   381
| "decr (NOT p) = NOT (decr p)" 
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   382
| "decr (And p q) = And (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   383
| "decr (Or p q) = Or (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   384
| "decr (Imp p q) = Imp (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   385
| "decr (Iff p q) = Iff (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   386
| "decr p = p"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   387
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   388
lemma decrnum: assumes nb: "numbound0 t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   389
  shows "Inum (x#bs) t = Inum bs (decrnum t)"
41849
1a65b780bd56 Some cleaning up
nipkow
parents: 41839
diff changeset
   390
  using nb by (induct t rule: decrnum.induct, simp_all)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   391
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   392
lemma decr: assumes nb: "bound0 p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   393
  shows "Ifm (x#bs) p = Ifm bs (decr p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   394
  using nb 
41849
1a65b780bd56 Some cleaning up
nipkow
parents: 41839
diff changeset
   395
  by (induct p rule: decr.induct, simp_all add: decrnum)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   396
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   397
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   398
by (induct p, simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   399
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   400
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   401
  "isatom T = True"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   402
| "isatom F = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   403
| "isatom (Lt a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   404
| "isatom (Le a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   405
| "isatom (Gt a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   406
| "isatom (Ge a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   407
| "isatom (Eq a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   408
| "isatom (NEq a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   409
| "isatom (Dvd i b) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   410
| "isatom (NDvd i b) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   411
| "isatom p = False"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   412
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   413
lemma numsubst0_numbound0: assumes nb: "numbound0 t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   414
  shows "numbound0 (numsubst0 t a)"
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   415
using nb by (induct a, auto)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   416
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   417
lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   418
  shows "bound0 (subst0 t p)"
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   419
using qf numsubst0_numbound0[OF nb] by (induct p, auto)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   420
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   421
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   422
by (induct p, simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   423
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   424
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   425
definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   426
  "djf f p q = (if q=T then T else if q=F then f p else 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   427
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q))"
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   428
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   429
definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   430
  "evaldjf f ps = foldr (djf f) ps F"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   431
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   432
lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   433
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   434
(cases "f p", simp_all add: Let_def djf_def) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   435
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   436
lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   437
  by(induct ps, simp_all add: evaldjf_def djf_Or)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   438
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   439
lemma evaldjf_bound0: 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   440
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   441
  shows "bound0 (evaldjf f xs)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   442
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   443
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   444
lemma evaldjf_qf: 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   445
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   446
  shows "qfree (evaldjf f xs)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   447
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   448
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   449
fun disjuncts :: "fm \<Rightarrow> fm list" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   450
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   451
| "disjuncts F = []"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   452
| "disjuncts p = [p]"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   453
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   454
fun conjuncts :: "fm \<Rightarrow> fm list" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   455
  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   456
| "conjuncts T = []"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   457
| "conjuncts p = [p]"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   458
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   459
lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   460
by(induct p rule: conjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   461
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   462
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   463
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   464
  assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   465
  hence "list_all qfree (disjuncts p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   466
    by (induct p rule: disjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   467
  thus ?thesis by (simp only: list_all_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   468
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   469
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   470
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   471
  assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   472
  hence "list_all qfree (conjuncts p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   473
    by (induct p rule: conjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   474
  thus ?thesis by (simp only: list_all_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   475
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   476
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   477
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   478
  "DJ f p \<equiv> evaldjf f (disjuncts p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   479
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   480
lemma DJ: assumes fdj: "\<forall> p q. f (Or p q) = Or (f p) (f q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   481
  and fF: "f F = F"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   482
  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   483
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   484
  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   485
    by (simp add: DJ_def evaldjf_ex) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   486
  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   487
  finally show ?thesis .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   488
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   489
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   490
lemma DJ_qf: assumes 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   491
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   492
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   493
proof(clarify)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   494
  fix  p assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   495
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   496
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   497
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   498
  
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   499
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   500
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   501
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   502
lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   503
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   504
proof(clarify)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   505
  fix p::fm and bs
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   506
  assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   507
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   508
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   509
  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   510
    by (simp add: DJ_def evaldjf_ex)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   511
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   512
  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   513
  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   514
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   515
  (* Simplification *)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   516
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   517
  (* Algebraic simplifications for nums *)
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   518
fun bnds:: "num \<Rightarrow> nat list" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   519
  "bnds (Bound n) = [n]"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   520
| "bnds (CN n c a) = n#(bnds a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   521
| "bnds (Neg a) = bnds a"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   522
| "bnds (Add a b) = (bnds a)@(bnds b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   523
| "bnds (Sub a b) = (bnds a)@(bnds b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   524
| "bnds (Mul i a) = bnds a"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   525
| "bnds (Floor a) = bnds a"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   526
| "bnds (CF c a b) = (bnds a)@(bnds b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   527
| "bnds a = []"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   528
fun lex_ns:: "nat list \<Rightarrow> nat list \<Rightarrow> bool" where
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   529
  "lex_ns [] ms = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   530
| "lex_ns ns [] = False"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   531
| "lex_ns (n#ns) (m#ms) = (n<m \<or> ((n = m) \<and> lex_ns ns ms)) "
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   532
definition lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool" where
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   533
  "lex_bnd t s \<equiv> lex_ns (bnds t) (bnds s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   534
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   535
fun maxcoeff:: "num \<Rightarrow> int" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   536
  "maxcoeff (C i) = abs i"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   537
| "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   538
| "maxcoeff (CF c t s) = max (abs c) (maxcoeff s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   539
| "maxcoeff t = 1"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   540
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   541
lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   542
  apply (induct t rule: maxcoeff.induct, auto) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   543
  done
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   544
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   545
fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" where
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   546
  "numgcdh (C i) = (\<lambda>g. gcd i g)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   547
| "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   548
| "numgcdh (CF c s t) = (\<lambda>g. gcd c (numgcdh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   549
| "numgcdh t = (\<lambda>g. 1)"
23858
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   550
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   551
definition
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   552
  numgcd :: "num \<Rightarrow> int"
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   553
where
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   554
  numgcd_def: "numgcd t = numgcdh t (maxcoeff t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   555
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   556
fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   557
  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   558
| "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   559
| "reducecoeffh (CF c s t) = (\<lambda> g. CF (c div g)  s (reducecoeffh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   560
| "reducecoeffh t = (\<lambda>g. t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   561
23858
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   562
definition
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   563
  reducecoeff :: "num \<Rightarrow> num"
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   564
where
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   565
  reducecoeff_def: "reducecoeff t =
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   566
  (let g = numgcd t in 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   567
  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   568
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   569
fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   570
  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   571
| "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   572
| "dvdnumcoeff (CF c s t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   573
| "dvdnumcoeff t = (\<lambda>g. False)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   574
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   575
lemma dvdnumcoeff_trans: 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   576
  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   577
  shows "dvdnumcoeff t g"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   578
  using dgt' gdg 
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29823
diff changeset
   579
  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29823
diff changeset
   580
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29823
diff changeset
   581
declare dvd_trans [trans add]
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   582
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   583
lemma numgcd0:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   584
  assumes g0: "numgcd t = 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   585
  shows "Inum bs t = 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   586
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   587
  have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   588
    by (induct t rule: numgcdh.induct, auto)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   589
  thus ?thesis using g0[simplified numgcd_def] by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   590
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   591
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   592
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   593
  using gp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   594
  by (induct t rule: numgcdh.induct, auto)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   595
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   596
lemma numgcd_pos: "numgcd t \<ge>0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   597
  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   598
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   599
lemma reducecoeffh:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   600
  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   601
  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   602
  using gt
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   603
proof(induct t rule: reducecoeffh.induct) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   604
  case (1 i) hence gd: "g dvd i" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   605
  from gp have gnz: "g \<noteq> 0" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   606
  from assms 1 show ?case by (simp add: real_of_int_div[OF gnz gd])
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   607
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   608
  case (2 n c t)  hence gd: "g dvd c" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   609
  from gp have gnz: "g \<noteq> 0" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   610
  from assms 2 show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   611
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   612
  case (3 c s t)  hence gd: "g dvd c" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   613
  from gp have gnz: "g \<noteq> 0" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   614
  from assms 3 show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps) 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   615
qed (auto simp add: numgcd_def gp)
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   616
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   617
fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   618
  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   619
| "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   620
| "ismaxcoeff (CF c s t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   621
| "ismaxcoeff t = (\<lambda>x. True)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   622
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   623
lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   624
by (induct t rule: ismaxcoeff.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   625
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   626
lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   627
proof (induct t rule: maxcoeff.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   628
  case (2 n c t)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   629
  hence H:"ismaxcoeff t (maxcoeff t)" .
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   630
  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   631
  from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   632
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   633
  case (3 c t s) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   634
  hence H1:"ismaxcoeff s (maxcoeff s)" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   635
  have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   636
  from ismaxcoeff_mono[OF H1 thh1] show ?case by (simp add: le_maxI1)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   637
qed simp_all
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   638
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   639
lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   640
  apply (unfold gcd_int_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   641
  apply (cases "i = 0", simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   642
  apply (cases "j = 0", simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   643
  apply (cases "abs i = 1", simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   644
  apply (cases "abs j = 1", simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   645
  apply auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   646
  done
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   647
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   648
  by (induct t rule: numgcdh.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   649
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   650
lemma dvdnumcoeff_aux:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   651
  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   652
  shows "dvdnumcoeff t (numgcdh t m)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   653
using assms
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   654
proof(induct t rule: numgcdh.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   655
  case (2 n c t) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   656
  let ?g = "numgcdh t m"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   657
  from 2 have th:"gcd c ?g > 1" by simp
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27456
diff changeset
   658
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   659
  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   660
  moreover {assume "abs c > 1" and gp: "?g > 1" with 2
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   661
    have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   662
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   663
    from dvdnumcoeff_trans[OF th' th] have ?case by simp }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   664
  moreover {assume "abs c = 0 \<and> ?g > 1"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   665
    with 2 have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   666
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   667
    from dvdnumcoeff_trans[OF th' th] have ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   668
    hence ?case by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   669
  moreover {assume "abs c > 1" and g0:"?g = 0" 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   670
    from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   671
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   672
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   673
  case (3 c s t) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   674
  let ?g = "numgcdh t m"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   675
  from 3 have th:"gcd c ?g > 1" by simp
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27456
diff changeset
   676
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   677
  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   678
  moreover {assume "abs c > 1" and gp: "?g > 1" with 3
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   679
    have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   680
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   681
    from dvdnumcoeff_trans[OF th' th] have ?case by simp }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   682
  moreover {assume "abs c = 0 \<and> ?g > 1"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   683
    with 3 have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   684
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   685
    from dvdnumcoeff_trans[OF th' th] have ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   686
    hence ?case by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   687
  moreover {assume "abs c > 1" and g0:"?g = 0" 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   688
    from numgcdh0[OF g0] have "m=0". with 3 g0 have ?case by simp }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   689
  ultimately show ?case by blast
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   690
qed auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   691
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   692
lemma dvdnumcoeff_aux2:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   693
  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   694
  using assms 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   695
proof (simp add: numgcd_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   696
  let ?mc = "maxcoeff t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   697
  let ?g = "numgcdh t ?mc"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   698
  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   699
  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   700
  assume H: "numgcdh t ?mc > 1"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   701
  from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" .
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   702
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   703
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   704
lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   705
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   706
  let ?g = "numgcd t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   707
  have "?g \<ge> 0"  by (simp add: numgcd_pos)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   708
  hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   709
  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   710
  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   711
  moreover { assume g1:"?g > 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   712
    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   713
    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   714
      by (simp add: reducecoeff_def Let_def)} 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   715
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   716
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   717
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   718
lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   719
by (induct t rule: reducecoeffh.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   720
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   721
lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   722
using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   723
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   724
consts
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   725
  numadd:: "num \<times> num \<Rightarrow> num"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   726
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   727
recdef numadd "measure (\<lambda> (t,s). size t + size s)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   728
  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   729
  (if n1=n2 then 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   730
  (let c = c1 + c2
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   731
  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   732
  else if n1 \<le> n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   733
  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   734
  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   735
  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   736
  "numadd (CF c1 t1 r1,CF c2 t2 r2) = 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   737
   (if t1 = t2 then 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   738
    (let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   739
   else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   740
   else CF c2 t2 (numadd(CF c1 t1 r1,r2)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   741
  "numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   742
  "numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   743
  "numadd (C b1, C b2) = C (b1+b2)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   744
  "numadd (a,b) = Add a b"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   745
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   746
lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   747
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   748
 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29265
diff changeset
   749
  apply (case_tac "n1 = n2", simp_all add: algebra_simps)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   750
  apply (simp only: left_distrib[symmetric])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   751
 apply simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   752
apply (case_tac "lex_bnd t1 t2", simp_all)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   753
 apply (case_tac "c1+c2 = 0")
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29265
diff changeset
   754
  by (case_tac "t1 = t2", simp_all add: algebra_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   755
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   756
lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   757
by (induct t s rule: numadd.induct, auto simp add: Let_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   758
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   759
fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   760
  "nummul (C j) = (\<lambda> i. C (i*j))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   761
| "nummul (CN n c t) = (\<lambda> i. CN n (c*i) (nummul t i))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   762
| "nummul (CF c t s) = (\<lambda> i. CF (c*i) t (nummul s i))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   763
| "nummul (Mul c t) = (\<lambda> i. nummul t (i*c))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   764
| "nummul t = (\<lambda> i. Mul i t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   765
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   766
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29265
diff changeset
   767
by (induct t rule: nummul.induct, auto simp add: algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   768
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   769
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   770
by (induct t rule: nummul.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   771
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   772
definition numneg :: "num \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   773
  "numneg t \<equiv> nummul t (- 1)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   774
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   775
definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   776
  "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   777
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   778
lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   779
using numneg_def nummul by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   780
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   781
lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   782
using numneg_def by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   783
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   784
lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   785
using numsub_def by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   786
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   787
lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   788
using numsub_def by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   789
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   790
lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   791
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   792
  have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   793
  
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   794
  have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   795
  also have "\<dots>" by (simp add: isint_add cti si)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   796
  finally show ?thesis .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   797
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   798
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   799
fun split_int:: "num \<Rightarrow> num \<times> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   800
  "split_int (C c) = (C 0, C c)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   801
| "split_int (CN n c b) = 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   802
     (let (bv,bi) = split_int b 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   803
       in (CN n c bv, bi))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   804
| "split_int (CF c a b) = 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   805
     (let (bv,bi) = split_int b 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   806
       in (bv, CF c a bi))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   807
| "split_int a = (a,C 0)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   808
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   809
lemma split_int: "\<And>tv ti. split_int t = (tv,ti) \<Longrightarrow> (Inum bs (Add tv ti) = Inum bs t) \<and> isint ti bs"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   810
proof (induct t rule: split_int.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   811
  case (2 c n b tv ti)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   812
  let ?bv = "fst (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   813
  let ?bi = "snd (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   814
  have "split_int b = (?bv,?bi)" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   815
  with 2(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   816
  from 2(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def)
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   817
  from 2(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   818
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   819
  case (3 c a b tv ti) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   820
  let ?bv = "fst (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   821
  let ?bi = "snd (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   822
  have "split_int b = (?bv,?bi)" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   823
  with 3(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   824
  from 3(2) have tibi: "ti = CF c a ?bi"
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   825
    by (simp add: Let_def split_def)
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   826
  from 3(2) b[symmetric] bii show ?case
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   827
    by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29265
diff changeset
   828
qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   829
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   830
lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) "
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   831
  by (induct t rule: split_int.induct) (auto simp add: Let_def split_def)
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   832
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   833
definition numfloor:: "num \<Rightarrow> num"
23858
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   834
where
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   835
  "numfloor t = (let (tv,ti) = split_int t in 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   836
  (case tv of C i \<Rightarrow> numadd (tv,ti) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   837
  | _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   838
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   839
lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)")
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   840
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   841
  let ?tv = "fst (split_int t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   842
  let ?ti = "snd (split_int t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   843
  have tvti:"split_int t = (?tv,?ti)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   844
  {assume H: "\<forall> v. ?tv \<noteq> C v"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   845
    hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   846
      by (cases ?tv, auto simp add: numfloor_def Let_def split_def numadd)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   847
    from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   848
    hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   849
    also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   850
      by (simp,subst tii[simplified isint_iff, symmetric]) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   851
    also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   852
    finally have ?thesis using th1 by simp}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   853
  moreover {fix v assume H:"?tv = C v" 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   854
    from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   855
    hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   856
    also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   857
      by (simp,subst tii[simplified isint_iff, symmetric]) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   858
    also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   859
    finally have ?thesis by (simp add: H numfloor_def Let_def split_def numadd) }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   860
  ultimately show ?thesis by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   861
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   862
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   863
lemma numfloor_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numfloor t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   864
  using split_int_nb[where t="t"]
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   865
  by (cases "fst(split_int t)" , auto simp add: numfloor_def Let_def split_def  numadd_nb)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   866
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   867
function simpnum:: "num \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   868
  "simpnum (C j) = C j"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   869
| "simpnum (Bound n) = CN n 1 (C 0)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   870
| "simpnum (Neg t) = numneg (simpnum t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   871
| "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   872
| "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   873
| "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   874
| "simpnum (Floor t) = numfloor (simpnum t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   875
| "simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   876
| "simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   877
by pat_completeness auto
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   878
termination by (relation "measure num_size") auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   879
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   880
lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   881
by (induct t rule: simpnum.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   882
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   883
lemma simpnum_numbound0[simp]: 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   884
  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   885
by (induct t rule: simpnum.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   886
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   887
fun nozerocoeff:: "num \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   888
  "nozerocoeff (C c) = True"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   889
| "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   890
| "nozerocoeff (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   891
| "nozerocoeff (Mul c t) = (c\<noteq>0 \<and> nozerocoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   892
| "nozerocoeff t = True"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   893
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   894
lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   895
by (induct a b rule: numadd.induct,auto simp add: Let_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   896
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   897
lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   898
  by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   899
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   900
lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   901
by (simp add: numneg_def nummul_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   902
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   903
lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   904
by (simp add: numsub_def numneg_nz numadd_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   905
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   906
lemma split_int_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   907
by (induct t rule: split_int.induct,auto simp add: Let_def split_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   908
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   909
lemma numfloor_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (numfloor t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   910
by (simp add: numfloor_def Let_def split_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   911
(cases "fst (split_int t)", simp_all add: split_int_nz numadd_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   912
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   913
lemma simpnum_nz: "nozerocoeff (simpnum t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   914
by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz numfloor_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   915
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   916
lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   917
proof (induct t rule: maxcoeff.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   918
  case (2 n c t)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   919
  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   920
  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   921
  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   922
  with 2 show ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   923
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   924
  case (3 c s t) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   925
  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   926
  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   927
  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   928
  with 3 show ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   929
qed auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   930
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   931
lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   932
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   933
  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   934
  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   935
  from maxcoeff_nz[OF nz th] show ?thesis .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   936
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   937
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   938
definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   939
  "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   940
   (let t' = simpnum t ; g = numgcd t' in 
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   941
      if g > 1 then (let g' = gcd n g in 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   942
        if g' = 1 then (t',n) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   943
        else (reducecoeffh t' g', n div g')) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   944
      else (t',n))))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   945
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   946
lemma simp_num_pair_ci:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   947
  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   948
  (is "?lhs = ?rhs")
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   949
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   950
  let ?t' = "simpnum t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   951
  let ?g = "numgcd ?t'"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   952
  let ?g' = "gcd n ?g"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   953
  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   954
  moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   955
  { assume nnz: "n \<noteq> 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   956
    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   957
    moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   958
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   959
      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31730
diff changeset
   960
      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   961
      hence "?g'= 1 \<or> ?g' > 1" by arith
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   962
      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   963
      moreover {assume g'1:"?g'>1"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   964
        from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   965
        let ?tt = "reducecoeffh ?t' ?g'"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   966
        let ?t = "Inum bs ?tt"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   967
        have gpdg: "?g' dvd ?g" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   968
        have gpdd: "?g' dvd n" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   969
        have gpdgp: "?g' dvd ?g'" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   970
        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   971
        have th2:"real ?g' * ?t = Inum bs ?t'" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   972
        from nnz g1 g'1 have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   973
        also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   974
        also have "\<dots> = (Inum bs ?t' / real n)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   975
          using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   976
        finally have "?lhs = Inum bs t / real n" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   977
        then have ?thesis using nnz g1 g'1 by (simp add: simp_num_pair_def) }
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   978
      ultimately have ?thesis by blast }
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   979
    ultimately have ?thesis by blast }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   980
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   981
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   982
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   983
lemma simp_num_pair_l:
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   984
  assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   985
  shows "numbound0 t' \<and> n' >0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   986
proof-
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   987
  let ?t' = "simpnum t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   988
  let ?g = "numgcd ?t'"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   989
  let ?g' = "gcd n ?g"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   990
  { assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   991
  moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   992
  { assume nnz: "n \<noteq> 0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   993
    {assume "\<not> ?g > 1" hence ?thesis using assms by (auto simp add: Let_def simp_num_pair_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   994
    moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   995
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   996
      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31730
diff changeset
   997
      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   998
      hence "?g'= 1 \<or> ?g' > 1" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   999
      moreover {assume "?g'=1" hence ?thesis using assms g1 g0
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1000
          by (auto simp add: Let_def simp_num_pair_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1001
      moreover {assume g'1:"?g'>1"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1002
        have gpdg: "?g' dvd ?g" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1003
        have gpdd: "?g' dvd n" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1004
        have gpdgp: "?g' dvd ?g'" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1005
        from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1006
        from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1007
        have "n div ?g' >0" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1008
        hence ?thesis using assms g1 g'1
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1009
          by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)}
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1010
      ultimately have ?thesis by blast }
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1011
    ultimately have ?thesis by blast } 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1012
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1013
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1014
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1015
fun not:: "fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1016
  "not (NOT p) = p"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1017
| "not T = F"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1018
| "not F = T"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1019
| "not (Lt t) = Ge t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1020
| "not (Le t) = Gt t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1021
| "not (Gt t) = Le t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1022
| "not (Ge t) = Lt t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1023
| "not (Eq t) = NEq t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1024
| "not (NEq t) = Eq t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1025
| "not (Dvd i t) = NDvd i t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1026
| "not (NDvd i t) = Dvd i t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1027
| "not (And p q) = Or (not p) (not q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1028
| "not (Or p q) = And (not p) (not q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1029
| "not p = NOT p"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1030
lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1031
  by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1032
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1033
  by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1034
lemma not_nb[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1035
  by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1036
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1037
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1038
  "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 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1039
   if p = q then p else And p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1040
lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1041
  by (cases "p=F \<or> q=F", simp_all add: conj_def) (cases p, simp_all)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1042
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1043
lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1044
  using conj_def by auto 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1045
lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1046
  using conj_def by auto 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1047
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1048
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1049
  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1050
       else if p=q then p else Or p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1051
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1052
lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1053
  by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1054
lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1055
  using disj_def by auto 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1056
lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1057
  using disj_def by auto 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1058
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1059
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1060
  "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 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1061
    else Imp p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1062
lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1063
  by (cases "p=F \<or> q=T",simp_all add: imp_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1064
lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1065
  using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1066
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1067
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1068
  "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1069
       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 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1070
  Iff p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1071
lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1072
  by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1073
(cases "not p= q", auto simp add:not)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1074
lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1075
  by (unfold iff_def,cases "p=q", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1076
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1077
fun check_int:: "num \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1078
  "check_int (C i) = True"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1079
| "check_int (Floor t) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1080
| "check_int (Mul i t) = check_int t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1081
| "check_int (Add t s) = (check_int t \<and> check_int s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1082
| "check_int (Neg t) = check_int t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1083
| "check_int (CF c t s) = check_int s"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1084
| "check_int t = False"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1085
lemma check_int: "check_int t \<Longrightarrow> isint t bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1086
by (induct t, auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1087
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1088
lemma rdvd_left1_int: "real \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1089
  by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1090
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1091
lemma rdvd_reduce: 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1092
  assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1093
  shows "real (d::int) rdvd real (c::int)*t = (real (d div g) rdvd real (c div g)*t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1094
proof
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1095
  assume d: "real d rdvd real c * t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1096
  from d rdvd_def obtain k where k_def: "real c * t = real d* real (k::int)" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1097
  from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1098
  from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1099
  from k_def kd_def kc_def have "real g * real kc * t = real g * real kd * real k" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1100
  hence "real kc * t = real kd * real k" using gp by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1101
  hence th:"real kd rdvd real kc * t" using rdvd_def by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1102
  from kd_def gp have th':"kd = d div g" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1103
  from kc_def gp have "kc = c div g" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1104
  with th th' show "real (d div g) rdvd real (c div g) * t" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1105
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1106
  assume d: "real (d div g) rdvd real (c div g) * t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1107
  from gp have gnz: "g \<noteq> 0" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1108
  thus "real d rdvd real c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real (c div g) * t"] real_of_int_div[OF gnz gd] real_of_int_div[OF gnz gc] by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1109
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1110
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1111
definition simpdvd :: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1112
  "simpdvd d t \<equiv> 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1113
   (let g = numgcd t in 
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1114
      if g > 1 then (let g' = gcd d g in 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1115
        if g' = 1 then (d, t) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1116
        else (d div g',reducecoeffh t g')) 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1117
      else (d, t))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1118
lemma simpdvd: 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1119
  assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1120
  shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1121
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1122
  let ?g = "numgcd t"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1123
  let ?g' = "gcd d ?g"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1124
  {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1125
  moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1126
  {assume g1:"?g>1" hence g0: "?g > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1127
    from g1 dnz have gp0: "?g' \<noteq> 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31730
diff changeset
  1128
    hence g'p: "?g' > 0" using gcd_ge_0_int[where x="d" and y="numgcd t"] by arith
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1129
    hence "?g'= 1 \<or> ?g' > 1" by arith
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1130
    moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1131
    moreover {assume g'1:"?g'>1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1132
      from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" ..
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1133
      let ?tt = "reducecoeffh t ?g'"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1134
      let ?t = "Inum bs ?tt"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1135
      have gpdg: "?g' dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1136
      have gpdd: "?g' dvd d" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1137
      have gpdgp: "?g' dvd ?g'" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1138
      from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1139
      have th2:"real ?g' * ?t = Inum bs t" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1140
      from assms g1 g0 g'1
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1141
      have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1142
        by (simp add: simpdvd_def Let_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1143
      also have "\<dots> = (real d rdvd (Inum bs t))"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1144
        using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified zdiv_self[OF gp0]] 
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1145
          th2[symmetric] by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1146
      finally have ?thesis by simp  }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1147
    ultimately have ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1148
  }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1149
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1150
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1151
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1152
function (sequential) simpfm :: "fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1153
  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1154
| "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1155
| "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1156
| "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1157
| "simpfm (NOT p) = not (simpfm p)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1158
| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1159
  | _ \<Rightarrow> Lt (reducecoeff a'))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1160
| "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1161
| "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1162
| "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1163
| "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1164
| "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1165
| "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1166
             else if (abs i = 1) \<and> check_int a then T
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1167
             else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1168
| "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1169
             else if (abs i = 1) \<and> check_int a then F
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1170
             else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1171
| "simpfm p = p"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1172
by pat_completeness auto
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1173
termination by (relation "measure fmsize") auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1174
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1175
lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1176
proof(induct p rule: simpfm.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1177
  case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1178
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1179
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1180
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1181
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1182
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1183
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1184
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1185
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1186
    hence gp: "real ?g > 0" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1187
    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1188
    with sa have "Inum bs a < 0 = (real ?g * ?r < real ?g * 0)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1189
    also have "\<dots> = (?r < 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1190
      by (simp only: mult_less_cancel_left) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1191
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1192
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1193
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1194
  case (7 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1195
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1196
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1197
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1198
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1199
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1200
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1201
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1202
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1203
    hence gp: "real ?g > 0" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1204
    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1205
    with sa have "Inum bs a \<le> 0 = (real ?g * ?r \<le> real ?g * 0)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1206
    also have "\<dots> = (?r \<le> 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1207
      by (simp only: mult_le_cancel_left) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1208
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1209
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1210
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1211
  case (8 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1212
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1213
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1214
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1215
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1216
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1217
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1218
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1219
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1220
    hence gp: "real ?g > 0" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1221
    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1222
    with sa have "Inum bs a > 0 = (real ?g * ?r > real ?g * 0)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1223
    also have "\<dots> = (?r > 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1224
      by (simp only: mult_less_cancel_left) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1225
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1226
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1227
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1228
  case (9 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1229
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1230
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1231
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1232
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1233
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1234
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1235
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1236
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1237
    hence gp: "real ?g > 0" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1238
    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1239
    with sa have "Inum bs a \<ge> 0 = (real ?g * ?r \<ge> real ?g * 0)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1240
    also have "\<dots> = (?r \<ge> 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1241
      by (simp only: mult_le_cancel_left) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1242
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1243
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1244
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1245
  case (10 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1246
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1247
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1248
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1249
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1250
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1251
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1252
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1253
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1254
    hence gp: "real ?g > 0" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1255
    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1256
    with sa have "Inum bs a = 0 = (real ?g * ?r = 0)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1257
    also have "\<dots> = (?r = 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1258
      by (simp add: mult_eq_0_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1259
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1260
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1261
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1262
  case (11 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1263
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1264
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1265
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1266
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1267
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1268
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1269
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}