src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
author wenzelm
Sat, 16 Apr 2011 20:49:48 +0200
changeset 42368 3b8498ac2314
parent 42364 8c674b3b8e44
child 42814 5af15f1e2ef6
permissions -rw-r--r--
proper subgoal addressing via SUBGOAL/CSUBGOAL -- assuming these tactics did not handle Subscript in any special way; tuned;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     1
(*  Title:      HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     2
    Author:     Amine Chaieb
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     3
*)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     4
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     5
header{* A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008 *}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     6
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
     7
theory Parametric_Ferrante_Rackoff
41849
1a65b780bd56 Some cleaning up
nipkow
parents: 41842
diff changeset
     8
imports Reflected_Multivariate_Polynomial Dense_Linear_Order DP_Library
41413
64cd30d6b0b8 explicit file specifications -- avoid secondary load path;
wenzelm
parents: 39246
diff changeset
     9
  "~~/src/HOL/Library/Efficient_Nat"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    10
begin
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    11
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    12
subsection {* Terms *}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    13
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    14
datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    15
  | Neg tm | Sub tm tm | CNP nat poly tm
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    16
  (* A size for poly to make inductive proofs simpler*)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    17
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    18
primrec tmsize :: "tm \<Rightarrow> nat" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    19
  "tmsize (CP c) = polysize c"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    20
| "tmsize (Bound n) = 1"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    21
| "tmsize (Neg a) = 1 + tmsize a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    22
| "tmsize (Add a b) = 1 + tmsize a + tmsize b"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    23
| "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    24
| "tmsize (Mul c a) = 1 + polysize c + tmsize a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    25
| "tmsize (CNP n c a) = 3 + polysize c + tmsize a "
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    26
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    27
  (* Semantics of terms tm *)
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    28
primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    29
  "Itm vs bs (CP c) = (Ipoly vs c)"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    30
| "Itm vs bs (Bound n) = bs!n"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    31
| "Itm vs bs (Neg a) = -(Itm vs bs a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    32
| "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    33
| "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    34
| "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    35
| "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"   
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    36
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    37
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    38
fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    39
  "allpolys P (CP c) = P c"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    40
| "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    41
| "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    42
| "allpolys P (Neg p) = allpolys P p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    43
| "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    44
| "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    45
| "allpolys P p = True"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    46
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    47
primrec tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    48
  "tmboundslt n (CP c) = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    49
| "tmboundslt n (Bound m) = (m < n)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    50
| "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    51
| "tmboundslt n (Neg a) = tmboundslt n a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    52
| "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    53
| "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    54
| "tmboundslt n (Mul i a) = tmboundslt n a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    55
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    56
primrec tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    57
  "tmbound0 (CP c) = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    58
| "tmbound0 (Bound n) = (n>0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    59
| "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    60
| "tmbound0 (Neg a) = tmbound0 a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    61
| "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    62
| "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    63
| "tmbound0 (Mul i a) = tmbound0 a"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    64
lemma tmbound0_I:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    65
  assumes nb: "tmbound0 a"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    66
  shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    67
using nb
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
    68
by (induct a rule: tm.induct,auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    69
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    70
primrec tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    71
  "tmbound n (CP c) = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    72
| "tmbound n (Bound m) = (n \<noteq> m)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    73
| "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    74
| "tmbound n (Neg a) = tmbound n a"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    75
| "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    76
| "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    77
| "tmbound n (Mul i a) = tmbound n a"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    78
lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    79
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    80
lemma tmbound_I: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    81
  assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    82
  shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    83
  using nb le bnd
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    84
  by (induct t rule: tm.induct , auto)
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    85
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    86
fun decrtm0:: "tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    87
  "decrtm0 (Bound n) = Bound (n - 1)"
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    88
| "decrtm0 (Neg a) = Neg (decrtm0 a)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    89
| "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    90
| "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    91
| "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    92
| "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    93
| "decrtm0 a = a"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
    94
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    95
fun incrtm0:: "tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
    96
  "incrtm0 (Bound n) = Bound (n + 1)"
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    97
| "incrtm0 (Neg a) = Neg (incrtm0 a)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    98
| "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
    99
| "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   100
| "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   101
| "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   102
| "incrtm0 a = a"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   103
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   104
lemma decrtm0: assumes nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   105
  shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   106
  using nb by (induct t rule: decrtm0.induct, simp_all)
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   107
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   108
lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   109
  by (induct t rule: decrtm0.induct, simp_all)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   110
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   111
primrec decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   112
  "decrtm m (CP c) = (CP c)"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   113
| "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   114
| "decrtm m (Neg a) = Neg (decrtm m a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   115
| "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   116
| "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   117
| "decrtm m (Mul c a) = Mul c (decrtm m a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   118
| "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   119
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   120
primrec removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   121
  "removen n [] = []"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   122
| "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   123
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   124
lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   125
  by (induct xs arbitrary: n, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   126
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   127
lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   128
  by (induct xs arbitrary: n, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   129
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   130
lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   131
  by (induct xs arbitrary: n, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   132
lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   133
  else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   134
proof(induct xs arbitrary: n m)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   135
  case Nil thus ?case by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   136
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   137
  case (Cons x xs n m)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   138
  {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   139
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   140
  {assume nxs: "\<not> (n \<ge> length (x#xs))" 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   141
    {assume mln: "m < n" hence ?case using Cons by (cases m, auto)}
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   142
    moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   143
    {assume mln: "\<not> (m < n)" 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   144
      {assume mxs: "m \<le> length (x#xs)" hence ?case using Cons by (cases m, auto)}
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   145
      moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   146
      {assume mxs: "\<not> (m \<le> length (x#xs))" 
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   147
        have th: "length (removen n (x#xs)) = length xs" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   148
          using removen_length[where n="n" and xs="x#xs"] nxs by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   149
        with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   150
        hence "(removen n (x#xs))!m = [] ! (m - length xs)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   151
          using th nth_length_exceeds[OF mxs'] by auto
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   152
        hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   153
          by auto
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33212
diff changeset
   154
        hence ?case using nxs mln mxs by auto }
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   155
      ultimately have ?case by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   156
    }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   157
    ultimately have ?case by blast
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   158
  } ultimately show ?case by blast
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   159
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   160
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   161
lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   162
  and nle: "m \<le> length bs" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   163
  shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   164
  using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   165
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   166
primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   167
  "tmsubst0 t (CP c) = CP c"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   168
| "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   169
| "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   170
| "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   171
| "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   172
| "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   173
| "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   174
lemma tmsubst0:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   175
  shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   176
  by (induct a rule: tm.induct) auto
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   177
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   178
lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   179
  by (induct a rule: tm.induct) auto
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   180
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   181
primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   182
  "tmsubst n t (CP c) = CP c"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   183
| "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   184
| "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a) 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   185
             else CNP m c (tmsubst n t a))"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   186
| "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   187
| "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   188
| "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   189
| "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   190
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   191
lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   192
  shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   193
using nb nlt
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   194
by (induct a rule: tm.induct,auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   195
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   196
lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   197
shows "tmbound0 (tmsubst 0 t a)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   198
using tnb
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   199
by (induct a rule: tm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   200
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   201
lemma tmsubst_nb: assumes tnb: "tmbound m t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   202
shows "tmbound m (tmsubst m t a)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   203
using tnb
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   204
by (induct a rule: tm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   205
lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   206
  by (induct t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   207
  (* Simplification *)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   208
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   209
consts
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   210
  tmadd:: "tm \<times> tm \<Rightarrow> tm"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   211
recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   212
  "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   213
  (if n1=n2 then 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   214
  (let c = c1 +\<^sub>p c2
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   215
  in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   216
  else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   217
  else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   218
  "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   219
  "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   220
  "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   221
  "tmadd (a,b) = Add a b"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   222
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   223
lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   224
apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   225
apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   226
apply (case_tac "n1 = n2", simp_all add: field_simps)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   227
apply (simp only: right_distrib[symmetric]) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   228
by (auto simp del: polyadd simp add: polyadd[symmetric])
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   229
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   230
lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   231
by (induct t s rule: tmadd.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   232
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   233
lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   234
by (induct t s rule: tmadd.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   235
lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   236
by (induct t s rule: tmadd.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   237
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   238
lemma tmadd_allpolys_npoly[simp]: "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmadd(t,s))" by (induct t s rule: tmadd.induct, simp_all add: Let_def polyadd_norm)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   239
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   240
fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   241
  "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   242
| "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   243
| "tmmul t = (\<lambda> i. Mul i t)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   244
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   245
lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   246
by (induct t arbitrary: i rule: tmmul.induct, simp_all add: field_simps)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   247
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   248
lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   249
by (induct t arbitrary: i rule: tmmul.induct, auto )
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   250
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   251
lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   252
by (induct t arbitrary: n rule: tmmul.induct, auto )
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   253
lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   254
by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   255
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   256
lemma tmmul_allpolys_npoly[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   257
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   258
  shows "allpolys isnpoly t \<Longrightarrow> isnpoly c \<Longrightarrow> allpolys isnpoly (tmmul t c)" by (induct t rule: tmmul.induct, simp_all add: Let_def polymul_norm)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   259
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   260
definition tmneg :: "tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   261
  "tmneg t \<equiv> tmmul t (C (- 1,1))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   262
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   263
definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   264
  "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   265
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   266
lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   267
using tmneg_def[of t] 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   268
apply simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   269
apply (subst number_of_Min)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   270
apply (simp only: of_int_minus)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   271
apply simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   272
done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   273
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   274
lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   275
using tmneg_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   276
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   277
lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   278
using tmneg_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   279
lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   280
using tmneg_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   281
lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   282
lemma tmneg_allpolys_npoly[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   283
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   284
  shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   285
  unfolding tmneg_def by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   286
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   287
lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   288
using tmsub_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   289
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   290
lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   291
using tmsub_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   292
lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   293
using tmsub_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   294
lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   295
using tmsub_def by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   296
lemma tmsub_allpolys_npoly[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   297
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   298
  shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   299
  unfolding tmsub_def by (simp add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   300
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   301
fun simptm:: "tm \<Rightarrow> tm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   302
  "simptm (CP j) = CP (polynate j)"
41821
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   303
| "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   304
| "simptm (Neg t) = tmneg (simptm t)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   305
| "simptm (Add t s) = tmadd (simptm t,simptm s)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   306
| "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   307
| "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
c118ae98dfbf recdef -> fun
krauss
parents: 41816
diff changeset
   308
| "simptm (CNP n c t) = (let c' = polynate c in if c' = 0\<^sub>p then simptm t else tmadd (CNP n c' (CP 0\<^sub>p ), simptm t))"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   309
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   310
lemma polynate_stupid: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   311
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   312
  shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a::{field_char_0, field_inverse_zero})" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   313
apply (subst polynate[symmetric])
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   314
apply simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   315
done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   316
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   317
lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   318
by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   319
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   320
lemma simptm_tmbound0[simp]: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   321
  "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   322
by (induct t rule: simptm.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   323
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   324
lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   325
by (induct t rule: simptm.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   326
lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   327
by (induct t rule: simptm.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   328
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   329
lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   330
  by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   331
lemma simptm_allpolys_npoly[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   332
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   333
  shows "allpolys isnpoly (simptm p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   334
  by (induct p rule: simptm.induct, auto simp add: Let_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   335
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   336
declare let_cong[fundef_cong del]
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   337
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   338
fun split0 :: "tm \<Rightarrow> (poly \<times> tm)" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   339
  "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   340
| "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   341
| "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   342
| "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   343
| "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   344
| "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   345
| "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   346
| "split0 t = (0\<^sub>p, t)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   347
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   348
declare let_cong[fundef_cong]
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   349
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   350
lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   351
  apply (rule exI[where x="fst (split0 p)"])
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   352
  apply (rule exI[where x="snd (split0 p)"])
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   353
  by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   354
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   355
lemma split0:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   356
  "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   357
  apply (induct t rule: split0.induct)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   358
  apply simp
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   359
  apply (simp add: Let_def split_def field_simps)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   360
  apply (simp add: Let_def split_def field_simps)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   361
  apply (simp add: Let_def split_def field_simps)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   362
  apply (simp add: Let_def split_def field_simps)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   363
  apply (simp add: Let_def split_def field_simps)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   364
  apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   365
  apply (simp add: Let_def split_def field_simps)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 35625
diff changeset
   366
  apply (simp add: Let_def split_def field_simps)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   367
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   368
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   369
lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   370
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   371
  fix c' t'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   372
  assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   373
  with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   374
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   375
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   376
lemma split0_nb0: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   377
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   378
  shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   379
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   380
  fix c' t'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   381
  assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   382
  with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   383
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   384
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   385
lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   386
  shows "tmbound0 (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   387
  using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   388
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   389
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   390
lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   391
  using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   392
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   393
lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   394
  using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   395
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   396
lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   397
 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   398
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   399
lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   400
by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   401
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   402
lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   403
 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   404
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   405
lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   406
by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   407
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   408
lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   409
  shows 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   410
  "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   411
  by (induct p rule: split0.induct, 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   412
    auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   413
    Let_def split_def split0_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   414
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   415
subsection{* Formulae *}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   416
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   417
datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   418
  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   419
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   420
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   421
  (* A size for fm *)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   422
fun fmsize :: "fm \<Rightarrow> nat" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   423
  "fmsize (NOT p) = 1 + fmsize p"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   424
| "fmsize (And p q) = 1 + fmsize p + fmsize q"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   425
| "fmsize (Or p q) = 1 + fmsize p + fmsize q"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   426
| "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   427
| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   428
| "fmsize (E p) = 1 + fmsize p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   429
| "fmsize (A p) = 4+ fmsize p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   430
| "fmsize p = 1"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   431
  (* several lemmas about fmsize *)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   432
lemma fmsize_pos[termination_simp]: "fmsize p > 0"        
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   433
by (induct p rule: fmsize.induct) simp_all
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   434
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   435
  (* Semantics of formulae (fm) *)
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   436
primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   437
  "Ifm vs bs T = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   438
| "Ifm vs bs F = False"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   439
| "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   440
| "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   441
| "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   442
| "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   443
| "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   444
| "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   445
| "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   446
| "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   447
| "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   448
| "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   449
| "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   450
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   451
fun not:: "fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   452
  "not (NOT (NOT p)) = not p"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   453
| "not (NOT p) = p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   454
| "not T = F"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   455
| "not F = T"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   456
| "not (Lt t) = Le (tmneg t)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   457
| "not (Le t) = Lt (tmneg t)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   458
| "not (Eq t) = NEq t"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   459
| "not (NEq t) = Eq t"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   460
| "not p = NOT p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   461
lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   462
by (induct p rule: not.induct) auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   463
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   464
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   465
  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   466
   if p = q then p else And p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   467
lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   468
by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   469
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   470
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   471
  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   472
       else if p=q then p else Or p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   473
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   474
lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   475
by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   476
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   477
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   478
  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   479
    else Imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   480
lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   481
by (cases "p=F \<or> q=T",simp_all add: imp_def) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   482
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   483
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   484
  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   485
       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   486
  Iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   487
lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   488
  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   489
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   490
  (* Quantifier freeness *)
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   491
fun qfree:: "fm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   492
  "qfree (E p) = False"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   493
| "qfree (A p) = False"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   494
| "qfree (NOT p) = qfree p" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   495
| "qfree (And p q) = (qfree p \<and> qfree q)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   496
| "qfree (Or  p q) = (qfree p \<and> qfree q)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   497
| "qfree (Imp p q) = (qfree p \<and> qfree q)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   498
| "qfree (Iff p q) = (qfree p \<and> qfree q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   499
| "qfree p = True"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   500
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   501
  (* Boundedness and substitution *)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   502
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   503
primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   504
  "boundslt n T = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   505
| "boundslt n F = True"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   506
| "boundslt n (Lt t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   507
| "boundslt n (Le t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   508
| "boundslt n (Eq t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   509
| "boundslt n (NEq t) = (tmboundslt n t)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   510
| "boundslt n (NOT p) = boundslt n p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   511
| "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   512
| "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   513
| "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   514
| "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   515
| "boundslt n (E p) = boundslt (Suc n) p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   516
| "boundslt n (A p) = boundslt (Suc n) p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   517
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   518
fun bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   519
  "bound0 T = True"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   520
| "bound0 F = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   521
| "bound0 (Lt a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   522
| "bound0 (Le a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   523
| "bound0 (Eq a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   524
| "bound0 (NEq a) = tmbound0 a"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   525
| "bound0 (NOT p) = bound0 p"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   526
| "bound0 (And p q) = (bound0 p \<and> bound0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   527
| "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   528
| "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   529
| "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   530
| "bound0 p = False"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   531
lemma bound0_I:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   532
  assumes bp: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   533
  shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   534
using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
41842
d8f76db6a207 added simp lemma nth_Cons_pos to List
nipkow
parents: 41838
diff changeset
   535
by (induct p rule: bound0.induct,auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   536
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   537
primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   538
  "bound m T = True"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   539
| "bound m F = True"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   540
| "bound m (Lt t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   541
| "bound m (Le t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   542
| "bound m (Eq t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   543
| "bound m (NEq t) = tmbound m t"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   544
| "bound m (NOT p) = bound m p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   545
| "bound m (And p q) = (bound m p \<and> bound m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   546
| "bound m (Or p q) = (bound m p \<and> bound m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   547
| "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   548
| "bound m (Iff p q) = (bound m p \<and> bound m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   549
| "bound m (E p) = bound (Suc m) p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   550
| "bound m (A p) = bound (Suc m) p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   551
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   552
lemma bound_I:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   553
  assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   554
  shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   555
  using bnd nb le tmbound_I[where bs=bs and vs = vs]
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   556
proof(induct p arbitrary: bs n rule: fm.induct)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   557
  case (E p bs n) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   558
  {fix y
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   559
    from E have bnd: "boundslt (length (y#bs)) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   560
      and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   561
    from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   562
  thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   563
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   564
  case (A p bs n) {fix y
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   565
    from A have bnd: "boundslt (length (y#bs)) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   566
      and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   567
    from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   568
  thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   569
qed auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   570
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   571
fun decr0 :: "fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   572
  "decr0 (Lt a) = Lt (decrtm0 a)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   573
| "decr0 (Le a) = Le (decrtm0 a)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   574
| "decr0 (Eq a) = Eq (decrtm0 a)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   575
| "decr0 (NEq a) = NEq (decrtm0 a)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   576
| "decr0 (NOT p) = NOT (decr0 p)" 
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   577
| "decr0 (And p q) = conj (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   578
| "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   579
| "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   580
| "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   581
| "decr0 p = p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   582
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   583
lemma decr0: assumes nb: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   584
  shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   585
  using nb 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   586
  by (induct p rule: decr0.induct, simp_all add: decrtm0)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   587
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   588
primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   589
  "decr m T = T"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   590
| "decr m F = F"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   591
| "decr m (Lt t) = (Lt (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   592
| "decr m (Le t) = (Le (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   593
| "decr m (Eq t) = (Eq (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   594
| "decr m (NEq t) = (NEq (decrtm m t))"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   595
| "decr m (NOT p) = NOT (decr m p)" 
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   596
| "decr m (And p q) = conj (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   597
| "decr m (Or p q) = disj (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   598
| "decr m (Imp p q) = imp (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   599
| "decr m (Iff p q) = iff (decr m p) (decr m q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   600
| "decr m (E p) = E (decr (Suc m) p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   601
| "decr m (A p) = A (decr (Suc m) p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   602
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   603
lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   604
  and nle: "m < length bs" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   605
  shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   606
  using bnd nb nle
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   607
proof(induct p arbitrary: bs m rule: fm.induct)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   608
  case (E p bs m) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   609
  {fix x
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   610
    from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   611
  and nle: "Suc m < length (x#bs)" by auto
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   612
    from E(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   613
  } thus ?case by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   614
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   615
  case (A p bs m)  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   616
  {fix x
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   617
    from A have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   618
  and nle: "Suc m < length (x#bs)" by auto
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   619
    from A(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   620
  } thus ?case by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   621
qed (auto simp add: decrtm removen_nth)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   622
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   623
primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   624
  "subst0 t T = T"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   625
| "subst0 t F = F"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   626
| "subst0 t (Lt a) = Lt (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   627
| "subst0 t (Le a) = Le (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   628
| "subst0 t (Eq a) = Eq (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   629
| "subst0 t (NEq a) = NEq (tmsubst0 t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   630
| "subst0 t (NOT p) = NOT (subst0 t p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   631
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   632
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   633
| "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   634
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   635
| "subst0 t (E p) = E p"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   636
| "subst0 t (A p) = A p"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   637
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   638
lemma subst0: assumes qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   639
  shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   640
using qf tmsubst0[where x="x" and bs="bs" and t="t"]
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   641
by (induct p rule: fm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   642
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   643
lemma subst0_nb:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   644
  assumes bp: "tmbound0 t" and qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   645
  shows "bound0 (subst0 t p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   646
using qf tmsubst0_nb[OF bp] bp
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   647
by (induct p rule: fm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   648
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   649
primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   650
  "subst n t T = T"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   651
| "subst n t F = F"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   652
| "subst n t (Lt a) = Lt (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   653
| "subst n t (Le a) = Le (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   654
| "subst n t (Eq a) = Eq (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   655
| "subst n t (NEq a) = NEq (tmsubst n t a)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   656
| "subst n t (NOT p) = NOT (subst n t p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   657
| "subst n t (And p q) = And (subst n t p) (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   658
| "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   659
| "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   660
| "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   661
| "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   662
| "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   663
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   664
lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   665
  shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   666
  using nb nlm
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   667
proof (induct p arbitrary: bs n t rule: fm.induct)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   668
  case (E p bs n) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   669
  {fix x 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   670
    from E have bn: "boundslt (length (x#bs)) p" by simp 
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   671
    from E have nlm: "Suc n \<le> length (x#bs)" by simp
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   672
    from E(1)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   673
    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   674
    by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   675
thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   676
next
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   677
  case (A p bs n)   
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   678
  {fix x 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   679
    from A have bn: "boundslt (length (x#bs)) p" by simp 
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   680
    from A have nlm: "Suc n \<le> length (x#bs)" by simp
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41413
diff changeset
   681
    from A(1)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp 
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   682
    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   683
    by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   684
thus ?case by simp 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   685
qed(auto simp add: tmsubst)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   686
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   687
lemma subst_nb: assumes tnb: "tmbound m t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   688
shows "bound m (subst m t p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   689
using tnb tmsubst_nb incrtm0_tmbound
39246
9e58f0499f57 modernized primrec
haftmann
parents: 38864
diff changeset
   690
by (induct p arbitrary: m t rule: fm.induct, auto)
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   691
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   692
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   693
by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   694
lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   695
by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   696
lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   697
by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   698
lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   699
 by (induct p rule: not.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   700
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   701
lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   702
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   703
lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   704
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   705
lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   706
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   707
lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   708
using conj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   709
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   710
lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   711
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   712
lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   713
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   714
lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   715
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   716
lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   717
using disj_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   718
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   719
lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   720
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   721
lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   722
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   723
lemma imp_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   724
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   725
lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   726
using imp_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   727
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   728
lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   729
  by (unfold iff_def,cases "p=q", auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   730
lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   731
using iff_def by (unfold iff_def,cases "p=q", auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   732
lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   733
using iff_def by (unfold iff_def,cases "p=q", auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   734
lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   735
using iff_def by auto 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   736
lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   737
by (induct p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   738
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   739
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   740
  "isatom T = True"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   741
| "isatom F = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   742
| "isatom (Lt a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   743
| "isatom (Le a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   744
| "isatom (Eq a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   745
| "isatom (NEq a) = True"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   746
| "isatom p = False"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   747
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   748
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   749
by (induct p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   750
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   751
definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   752
  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   753
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   754
definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   755
  "evaldjf f ps \<equiv> foldr (djf f) ps F"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   756
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   757
lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   758
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   759
(cases "f p", simp_all add: Let_def djf_def) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   760
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   761
lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   762
  by(induct ps, simp_all add: evaldjf_def djf_Or)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   763
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   764
lemma evaldjf_bound0: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   765
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   766
  shows "bound0 (evaldjf f xs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   767
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   768
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   769
lemma evaldjf_qf: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   770
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   771
  shows "qfree (evaldjf f xs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   772
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   773
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   774
fun disjuncts :: "fm \<Rightarrow> fm list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   775
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   776
| "disjuncts F = []"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   777
| "disjuncts p = [p]"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   778
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   779
lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   780
by(induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   781
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   782
lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   783
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   784
  assume nb: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   785
  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   786
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   787
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   788
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   789
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   790
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   791
  assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   792
  hence "list_all qfree (disjuncts p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   793
    by (induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   794
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   795
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   796
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   797
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   798
  "DJ f p \<equiv> evaldjf f (disjuncts p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   799
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   800
lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   801
  and fF: "f F = F"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   802
  shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   803
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   804
  have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   805
    by (simp add: DJ_def evaldjf_ex) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   806
  also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   807
  finally show ?thesis .
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   808
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   809
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   810
lemma DJ_qf: assumes 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   811
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   812
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   813
proof(clarify)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   814
  fix  p assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   815
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   816
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   817
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   818
  
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   819
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   820
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   821
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   822
lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   823
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   824
proof(clarify)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   825
  fix p::fm and bs
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   826
  assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   827
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   828
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   829
  have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   830
    by (simp add: DJ_def evaldjf_ex)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   831
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   832
  also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   833
  finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   834
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   835
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   836
fun conjuncts :: "fm \<Rightarrow> fm list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   837
  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   838
| "conjuncts T = []"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
   839
| "conjuncts p = [p]"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   840
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   841
definition list_conj :: "fm list \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   842
  "list_conj ps \<equiv> foldr conj ps T"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   843
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
   844
definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   845
  "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   846
                   in conj (decr0 (list_conj yes)) (f (list_conj no)))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   847
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   848
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   849
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   850
  assume qf: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   851
  hence "list_all qfree (conjuncts p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   852
    by (induct p rule: conjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   853
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   854
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   855
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   856
lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   857
by(induct p rule: conjuncts.induct, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   858
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   859
lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   860
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   861
  assume nb: "bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   862
  hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   863
  thus ?thesis by (simp only: list_all_iff)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   864
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   865
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   866
fun islin :: "fm \<Rightarrow> bool" where
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   867
  "islin (And p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   868
| "islin (Or p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   869
| "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   870
| "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   871
| "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   872
| "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   873
| "islin (NOT p) = False"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   874
| "islin (Imp p q) = False"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   875
| "islin (Iff p q) = False"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   876
| "islin p = bound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   877
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   878
lemma islin_stupid: assumes nb: "tmbound0 p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   879
  shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   880
  using nb by (cases p, auto, case_tac nat, auto)+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   881
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   882
definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   883
definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
38864
4abe644fcea5 formerly unnamed infix equality now named HOL.eq
haftmann
parents: 38795
diff changeset
   884
definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   885
definition "neq p = not (eq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   886
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   887
lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   888
  apply(simp add: lt_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   889
  apply(cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   890
  apply (case_tac poly, simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   891
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   892
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   893
lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   894
  apply(simp add: le_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   895
  apply(cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   896
  apply (case_tac poly, simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   897
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   898
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   899
lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   900
  apply(simp add: eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   901
  apply(cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   902
  apply (case_tac poly, simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   903
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   904
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   905
lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   906
  by(simp add: neq_def eq)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   907
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   908
lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   909
  apply (simp add: lt_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   910
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   911
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   912
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   913
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   914
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   915
lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   916
  apply (simp add: le_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   917
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   918
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   919
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   920
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   921
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   922
lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   923
  apply (simp add: eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   924
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   925
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   926
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   927
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   928
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   929
lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   930
  apply (simp add: neq_def eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   931
  apply (cases p, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   932
  apply (case_tac poly, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   933
  apply (case_tac nat, simp_all)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   934
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   935
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   936
definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   937
definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   938
definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   939
definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   940
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   941
lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   942
  shows "islin (simplt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   943
  unfolding simplt_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   944
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   945
by (auto simp add: lt_lin Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly])
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   946
  
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   947
lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   948
  shows "islin (simple t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   949
  unfolding simple_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   950
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   951
by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] le_lin)
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   952
lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   953
  shows "islin (simpeq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   954
  unfolding simpeq_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   955
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   956
by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   957
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   958
lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   959
  shows "islin (simpneq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   960
  unfolding simpneq_def 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   961
  using split0_nb0'
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   962
by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] neq_lin)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   963
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   964
lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   965
  by (cases "split0 s", auto)
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   966
lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   967
  and n: "allpolys isnpoly t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   968
  shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   969
  using n
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   970
  by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   971
lemma simplt[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   972
  shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   973
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   974
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   975
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   976
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   977
      using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   978
      by (simp add: simplt_def Let_def split_def lt)}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   979
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   980
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   981
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   982
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   983
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   984
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   985
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   986
lemma simple[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   987
  shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   988
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   989
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   990
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   991
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   992
      using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   993
      by (simp add: simple_def Let_def split_def le)}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   994
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   995
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   996
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   997
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   998
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
   999
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1000
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1001
lemma simpeq[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1002
  shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1003
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1004
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1005
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1006
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1007
      using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1008
      by (simp add: simpeq_def Let_def split_def)}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1009
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1010
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1011
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1012
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1013
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1014
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1015
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1016
lemma simpneq[simp]:
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1017
  shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1018
proof-
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1019
  have n: "allpolys isnpoly (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1020
  let ?t = "simptm t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1021
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1022
      using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1023
      by (simp add: simpneq_def Let_def split_def )}
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1024
  moreover
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1025
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1026
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1027
  }
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1028
  ultimately show ?thesis by blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1029
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1030
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1031
lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1032
  apply (simp add: lt_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1033
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1034
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1035
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1036
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1037
lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1038
  apply (simp add: le_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1039
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1040
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1041
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1042
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1043
lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1044
  apply (simp add: eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1045
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1046
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1047
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1048
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1049
lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1050
  apply (simp add: neq_def eq_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1051
  apply (cases t, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1052
  apply (case_tac poly, auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1053
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1054
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1055
lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1056
  shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1057
  using split0 [of "simptm t" vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1058
proof(simp add: simplt_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1059
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1060
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1061
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1062
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1063
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1064
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1065
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1066
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1067
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1068
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1069
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1070
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1071
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1072
lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1073
  shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1074
  using split0 [of "simptm t" vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1075
proof(simp add: simple_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1076
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1077
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1078
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1079
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1080
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1081
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1082
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1083
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1084
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1085
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1086
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1087
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1088
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1089
lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1090
  shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1091
  using split0 [of "simptm t" vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1092
proof(simp add: simpeq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1093
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1094
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1095
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1096
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1097
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1098
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1099
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1100
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1101
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1102
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1103
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1104
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1105
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1106
lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1107
  shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1108
  using split0 [of "simptm t" vs bs]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1109
proof(simp add: simpneq_def Let_def split_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1110
  assume nb: "tmbound0 t"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1111
  hence nb': "tmbound0 (simptm t)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1112
  let ?c = "fst (split0 (simptm t))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1113
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1114
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1115
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1116
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1117
  from iffD1[OF isnpolyh_unique[OF ths] th]
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1118
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1119
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1120
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1121
qed
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1122
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
  1123
fun conjs   :: "fm \<Rightarrow> fm list" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1124
  "conjs (And p q) = (conjs p)@(conjs q)"
41822
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
  1125
| "conjs T = []"
27afef7d6c37 recdef -> fun
krauss
parents: 41821
diff changeset
  1126
| "conjs p = [p]"
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1127
lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1128
by (induct p rule: conjs.induct, auto)
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
  1129
definition list_disj :: "fm list \<Rightarrow> fm" where
33152
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1130
  "list_disj ps \<equiv> foldr disj ps F"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1131
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1132
lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1133
  by (induct ps, auto simp add: list_conj_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1134
lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1135
  by (induct ps, auto simp add: list_conj_def conj_qf)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1136
lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1137
  by (induct ps, auto simp add: list_disj_def)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1138
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1139
lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1140
  unfolding conj_def by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1141
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1142
lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1143
  apply (induct p rule: conjs.induct) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1144
  apply (unfold conjs.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1145
  apply (unfold set_append)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1146
  apply (unfold ball_Un)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1147
  apply (unfold bound.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1148
  apply auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1149
  done
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1150
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1151
lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1152
  apply (induct p rule: conjs.induct) 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1153
  apply (unfold conjs.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1154
  apply (unfold set_append)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1155
  apply (unfold ball_Un)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1156
  apply (unfold boundslt.simps)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1157
  apply blast
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1158
by simp_all
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1159
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1160
lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1161
  unfolding list_conj_def
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1162
  by (induct ps, auto simp add: conj_boundslt)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1163
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1164
lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1165
  shows "bound n (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1166
  using bnd
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1167
  unfolding list_conj_def
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1168
  by (induct ps, auto simp add: conj_nb)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1169
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1170
lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1171
unfolding list_conj_def by (induct ps , auto)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1172
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1173
lemma CJNB_qe: 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1174
  assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1175
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1176
proof(clarify)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1177
  fix bs p
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1178
  assume qfp: "qfree p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1179
  let ?cjs = "conjuncts p"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1180
  let ?yes = "fst (partition bound0 ?cjs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1181
  let ?no = "snd (partition bound0 ?cjs)"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1182
  let ?cno = "list_conj ?no"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1183
  let ?cyes = "list_conj ?yes"
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1184
  have part: "partition bound0 ?cjs = (?yes,?no)" by simp
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1185
  from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1186
  hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1187
  hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1188
  from conjuncts_qf[OF qfp] partition_set[OF part] 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1189
  have " \<forall>q\<in> set ?no. qfree q" by auto
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1190
  hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1191
  with qe have cno_qf:"qfree (qe ?cno )" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1192
    and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1193
  from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
241cfaed158f Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff changeset
  1194
    by (simp add: CJNB_def Let_def conj_qf split_def)