src/HOL/Decision_Procs/MIR.thy
author nipkow
Fri, 05 Aug 2016 15:44:53 +0200
changeset 63600 d0fa16751d14
parent 62342 1cf129590be8
child 64240 eabf80376aab
permissions -rw-r--r--
fixed floor proof
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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57c68b3af2ea Updated paths in Decision_Procs comments and NEWS
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(*  Title:      HOL/Decision_Procs/MIR.thy
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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    Author:     Amine Chaieb
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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*)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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theory MIR
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imports Complex_Main Dense_Linear_Order DP_Library
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0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
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     7
  "~~/src/HOL/Library/Code_Target_Numeral" "~~/src/HOL/Library/Old_Recdef"
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9f90ac19e32b established Plain theory and image
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begin
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61586
5197a2ecb658 isabelle update_cartouches -c -t;
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section \<open>Quantifier elimination for \<open>\<real> (0, 1, +, floor, <)\<close>\<close>
27456
52c7c42e7e27 code antiquotation roaring ahead
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61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
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    12
declare of_int_floor_cancel [simp del]
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960b0ca9ae5d tuned proofs -- more structure, less warnings;
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lemma myle:
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  fixes a b :: "'a::{ordered_ab_group_add}"
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1a65b780bd56 Some cleaning up
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    16
  shows "(a \<le> b) = (0 \<le> b - a)"
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  by (metis add_0_left add_le_cancel_right diff_add_cancel)
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960b0ca9ae5d tuned proofs -- more structure, less warnings;
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lemma myless:
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  fixes a b :: "'a::{ordered_ab_group_add}"
41849
1a65b780bd56 Some cleaning up
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parents: 41839
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    21
  shows "(a < b) = (0 < b - a)"
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parents: 51272
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  by (metis le_iff_diff_le_0 less_le_not_le myle)
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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(* Periodicity of dvd *)
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paulson <lp15@cam.ac.uk>
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lemmas dvd_period = zdvd_period
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32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
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(* The Divisibility relation between reals *)
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definition rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50)
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77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
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  where "x rdvd y \<longleftrightarrow> (\<exists>k::int. y = x * real_of_int k)"
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lemma int_rdvd_real:
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  "real_of_int (i::int) rdvd x = (i dvd \<lfloor>x\<rfloor> \<and> real_of_int \<lfloor>x\<rfloor> = x)" (is "?l = ?r")
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proof
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  assume "?l"
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77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
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    35
  hence th: "\<exists> k. x=real_of_int (i*k)" by (simp add: rdvd_def)
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  hence th': "real_of_int \<lfloor>x\<rfloor> = x" by (auto simp del: of_int_mult)
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    37
  with th have "\<exists> k. real_of_int \<lfloor>x\<rfloor> = real_of_int (i*k)" by simp
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    38
  hence "\<exists>k. \<lfloor>x\<rfloor> = i*k" by presburger
f02b26f7d39d prefer symbols for "floor", "ceiling";
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    39
  thus ?r using th' by (simp add: dvd_def)
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next
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bdc1e2f0a86a eliminated \<Colon>;
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    41
  assume "?r" hence "(i::int) dvd \<lfloor>x::real\<rfloor>" ..
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  hence "\<exists>k. real_of_int \<lfloor>x\<rfloor> = real_of_int (i*k)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
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    43
    by (metis (no_types) dvd_def)
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1e7ccd864b62 isabelle update_cartouches;
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    44
  thus ?l using \<open>?r\<close> by (simp add: rdvd_def)
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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    46
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
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    47
lemma int_rdvd_iff: "(real_of_int (i::int) rdvd real_of_int t) = (i dvd t)"
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
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    48
  by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only: of_int_mult[symmetric])
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
    49
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
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    50
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lemma rdvd_abs1: "(\<bar>real_of_int d\<bar> rdvd t) = (real_of_int (d ::int) rdvd t)"
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proof
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
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    53
  assume d: "real_of_int d rdvd t"
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  from d int_rdvd_real have d2: "d dvd \<lfloor>t\<rfloor>" and ti: "real_of_int \<lfloor>t\<rfloor> = t"
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    by auto
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1135b8de26c3 more symbols;
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  from iffD2[OF abs_dvd_iff] d2 have "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>" by blast
1135b8de26c3 more symbols;
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    58
  with ti int_rdvd_real[symmetric] have "real_of_int \<bar>d\<bar> rdvd t" by blast
1135b8de26c3 more symbols;
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    59
  thus "\<bar>real_of_int d\<bar> rdvd t" by simp
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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next
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1135b8de26c3 more symbols;
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    61
  assume "\<bar>real_of_int d\<bar> rdvd t" hence "real_of_int \<bar>d\<bar> rdvd t" by simp
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    62
  with int_rdvd_real[where i="\<bar>d\<bar>" and x="t"]
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    63
  have d2: "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>" and ti: "real_of_int \<lfloor>t\<rfloor> = t"
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    64
    by auto
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f02b26f7d39d prefer symbols for "floor", "ceiling";
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diff changeset
    65
  from iffD1[OF abs_dvd_iff] d2 have "d dvd \<lfloor>t\<rfloor>" by blast
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
    66
  with ti int_rdvd_real[symmetric] show "real_of_int d rdvd t" by blast
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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    67
qed
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parents:
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    68
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
    69
lemma rdvd_minus: "(real_of_int (d::int) rdvd t) = (real_of_int d rdvd -t)"
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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    70
  apply (auto simp add: rdvd_def)
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
    71
  apply (rule_tac x="-k" in exI, simp)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    72
  apply (rule_tac x="-k" in exI, simp)
51369
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    73
  done
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
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    74
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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    75
lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)"
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    76
  by (auto simp add: rdvd_def)
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parents:
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    77
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
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    78
lemma rdvd_mult:
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324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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parents:
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    79
  assumes knz: "k\<noteq>0"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
    80
  shows "(real_of_int (n::int) * real_of_int (k::int) rdvd x * real_of_int k) = (real_of_int n rdvd x)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
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parents: 51272
diff changeset
    81
  using knz by (simp add: rdvd_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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parents:
diff changeset
    82
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    83
  (*********************************************************************************)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    84
  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    85
  (*********************************************************************************)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    86
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
    87
datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
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    88
  | Mul int num | Floor num| CF int num num
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    89
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    90
  (* A size for num to make inductive proofs simpler*)
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
    91
primrec num_size :: "num \<Rightarrow> nat" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    92
 "num_size (C c) = 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    93
| "num_size (Bound n) = 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    94
| "num_size (Neg a) = 1 + num_size a"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    95
| "num_size (Add a b) = 1 + num_size a + num_size b"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    96
| "num_size (Sub a b) = 3 + num_size a + num_size b"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    97
| "num_size (CN n c a) = 4 + num_size a "
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    98
| "num_size (CF c a b) = 4 + num_size a + num_size b"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
    99
| "num_size (Mul c a) = 1 + num_size a"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   100
| "num_size (Floor a) = 1 + num_size a"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   101
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   102
  (* Semantics of numeral terms (num) *)
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   103
primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   104
  "Inum bs (C c) = (real_of_int c)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   105
| "Inum bs (Bound n) = bs!n"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   106
| "Inum bs (CN n c a) = (real_of_int c) * (bs!n) + (Inum bs a)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   107
| "Inum bs (Neg a) = -(Inum bs a)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   108
| "Inum bs (Add a b) = Inum bs a + Inum bs b"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   109
| "Inum bs (Sub a b) = Inum bs a - Inum bs b"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   110
| "Inum bs (Mul c a) = (real_of_int c) * Inum bs a"
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   111
| "Inum bs (Floor a) = real_of_int \<lfloor>Inum bs a\<rfloor>"
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   112
| "Inum bs (CF c a b) = real_of_int c * real_of_int \<lfloor>Inum bs a\<rfloor> + Inum bs b"
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   113
definition "isint t bs \<equiv> real_of_int \<lfloor>Inum bs t\<rfloor> = Inum bs t"
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   114
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   115
lemma isint_iff: "isint n bs = (real_of_int \<lfloor>Inum bs n\<rfloor> = Inum bs n)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   116
  by (simp add: isint_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   117
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   118
lemma isint_Floor: "isint (Floor n) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   119
  by (simp add: isint_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   120
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   121
lemma isint_Mul: "isint e bs \<Longrightarrow> isint (Mul c e) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   122
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   123
  let ?e = "Inum bs e"
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   124
  assume be: "isint e bs" hence efe:"real_of_int \<lfloor>?e\<rfloor> = ?e" by (simp add: isint_iff)
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   125
  have "real_of_int \<lfloor>Inum bs (Mul c e)\<rfloor> = real_of_int \<lfloor>real_of_int (c * \<lfloor>?e\<rfloor>)\<rfloor>"
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   126
    using efe by simp
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   127
  also have "\<dots> = real_of_int (c* \<lfloor>?e\<rfloor>)" by (metis floor_of_int)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   128
  also have "\<dots> = real_of_int c * ?e" using efe by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   129
  finally show ?thesis using isint_iff by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   130
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   131
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   132
lemma isint_neg: "isint e bs \<Longrightarrow> isint (Neg e) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   133
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   134
  let ?I = "\<lambda> t. Inum bs t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   135
  assume ie: "isint e bs"
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   136
  hence th: "real_of_int \<lfloor>?I e\<rfloor> = ?I e" by (simp add: isint_def)
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   137
  have "real_of_int \<lfloor>?I (Neg e)\<rfloor> = real_of_int \<lfloor>- (real_of_int \<lfloor>?I e\<rfloor>)\<rfloor>"
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   138
    by (simp add: th)
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   139
  also have "\<dots> = - real_of_int \<lfloor>?I e\<rfloor>" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   140
  finally show "isint (Neg e) bs" by (simp add: isint_def th)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   141
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   142
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   143
lemma isint_sub:
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   144
  assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   145
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   146
  let ?I = "\<lambda> t. Inum bs t"
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   147
  from ie have th: "real_of_int \<lfloor>?I e\<rfloor> = ?I e" by (simp add: isint_def)
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   148
  have "real_of_int \<lfloor>?I (Sub (C c) e)\<rfloor> = real_of_int \<lfloor>real_of_int (c - \<lfloor>?I e\<rfloor>)\<rfloor>"
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   149
    by (simp add: th)
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   150
  also have "\<dots> = real_of_int (c - \<lfloor>?I e\<rfloor>)" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   151
  finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   152
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   153
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   154
lemma isint_add:
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   155
  assumes ai: "isint a bs" and bi: "isint b bs"
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   156
  shows "isint (Add a b) bs"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   157
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   158
  let ?a = "Inum bs a"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   159
  let ?b = "Inum bs b"
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   160
  from ai bi isint_iff have "real_of_int \<lfloor>?a + ?b\<rfloor> = real_of_int \<lfloor>real_of_int \<lfloor>?a\<rfloor> + real_of_int \<lfloor>?b\<rfloor>\<rfloor>"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   161
    by simp
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   162
  also have "\<dots> = real_of_int \<lfloor>?a\<rfloor> + real_of_int \<lfloor>?b\<rfloor>" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   163
  also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   164
  finally show "isint (Add a b) bs" by (simp add: isint_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   165
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   166
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   167
lemma isint_c: "isint (C j) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   168
  by (simp add: isint_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   169
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   170
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   171
    (* FORMULAE *)
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   172
datatype fm  =
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   173
  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   174
  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   175
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   176
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   177
  (* A size for fm *)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   178
fun fmsize :: "fm \<Rightarrow> nat" where
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   179
 "fmsize (NOT p) = 1 + fmsize p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   180
| "fmsize (And p q) = 1 + fmsize p + fmsize q"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   181
| "fmsize (Or p q) = 1 + fmsize p + fmsize q"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   182
| "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   183
| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   184
| "fmsize (E p) = 1 + fmsize p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   185
| "fmsize (A p) = 4+ fmsize p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   186
| "fmsize (Dvd i t) = 2"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   187
| "fmsize (NDvd i t) = 2"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   188
| "fmsize p = 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   189
  (* several lemmas about fmsize *)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   190
lemma fmsize_pos: "fmsize p > 0"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   191
  by (induct p rule: fmsize.induct) simp_all
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   192
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   193
  (* Semantics of formulae (fm) *)
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   194
primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   195
  "Ifm bs T = True"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   196
| "Ifm bs F = False"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   197
| "Ifm bs (Lt a) = (Inum bs a < 0)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   198
| "Ifm bs (Gt a) = (Inum bs a > 0)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   199
| "Ifm bs (Le a) = (Inum bs a \<le> 0)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   200
| "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   201
| "Ifm bs (Eq a) = (Inum bs a = 0)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   202
| "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   203
| "Ifm bs (Dvd i b) = (real_of_int i rdvd Inum bs b)"
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   204
| "Ifm bs (NDvd i b) = (\<not>(real_of_int i rdvd Inum bs b))"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   205
| "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   206
| "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   207
| "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   208
| "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   209
| "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   210
| "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   211
| "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   212
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   213
consts prep :: "fm \<Rightarrow> fm"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   214
recdef prep "measure fmsize"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   215
  "prep (E T) = T"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   216
  "prep (E F) = F"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   217
  "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   218
  "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   219
  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   220
  "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   221
  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   222
  "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   223
  "prep (E p) = E (prep p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   224
  "prep (A (And p q)) = And (prep (A p)) (prep (A q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   225
  "prep (A p) = prep (NOT (E (NOT p)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   226
  "prep (NOT (NOT p)) = prep p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   227
  "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   228
  "prep (NOT (A p)) = prep (E (NOT p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   229
  "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   230
  "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   231
  "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))"
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   232
  "prep (NOT p) = NOT (prep p)"
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   233
  "prep (Or p q) = Or (prep p) (prep q)"
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   234
  "prep (And p q) = And (prep p) (prep q)"
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   235
  "prep (Imp p q) = prep (Or (NOT p) q)"
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   236
  "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))"
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   237
  "prep p = p"
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   238
(hints simp add: fmsize_pos)
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   239
lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
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  by (induct p rule: prep.induct) auto
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   241
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   242
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   243
  (* Quantifier freeness *)
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fun qfree:: "fm \<Rightarrow> bool" where
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  "qfree (E p) = False"
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  | "qfree (A p) = False"
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parents: 61652
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   247
  | "qfree (NOT p) = qfree p"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
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parents: 61652
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   248
  | "qfree (And p q) = (qfree p \<and> qfree q)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
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parents: 61652
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   249
  | "qfree (Or  p q) = (qfree p \<and> qfree q)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
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   250
  | "qfree (Imp p q) = (qfree p \<and> qfree q)"
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   251
  | "qfree (Iff p q) = (qfree p \<and> qfree q)"
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  | "qfree p = True"
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   253
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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  (* Boundedness and substitution *)
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   255
primrec numbound0 :: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
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parents:
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   256
  "numbound0 (C c) = True"
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   257
  | "numbound0 (Bound n) = (n>0)"
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   258
  | "numbound0 (CN n i a) = (n > 0 \<and> numbound0 a)"
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   259
  | "numbound0 (Neg a) = numbound0 a"
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   260
  | "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
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parents: 61652
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   261
  | "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)"
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   262
  | "numbound0 (Mul i a) = numbound0 a"
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   263
  | "numbound0 (Floor a) = numbound0 a"
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parents: 61652
diff changeset
   264
  | "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)"
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diff changeset
   265
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   266
lemma numbound0_I:
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   267
  assumes nb: "numbound0 a"
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   268
  shows "Inum (b#bs) a = Inum (b'#bs) a"
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   269
  using nb by (induct a) auto
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diff changeset
   270
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paulson <lp15@cam.ac.uk>
parents: 61652
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   271
lemma numbound0_gen:
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   272
  assumes nb: "numbound0 t" and ti: "isint t (x#bs)"
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   273
  shows "\<forall> y. isint t (y#bs)"
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6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   274
  using nb ti
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   275
proof(clarify)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   276
  fix y
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   277
  from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def]
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   278
  show "isint t (y#bs)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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diff changeset
   279
    by (simp add: isint_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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diff changeset
   280
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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   281
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   282
primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
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chaieb
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   283
  "bound0 T = True"
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   284
  | "bound0 F = True"
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   285
  | "bound0 (Lt a) = numbound0 a"
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   286
  | "bound0 (Le a) = numbound0 a"
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   287
  | "bound0 (Gt a) = numbound0 a"
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   288
  | "bound0 (Ge a) = numbound0 a"
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   289
  | "bound0 (Eq a) = numbound0 a"
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   290
  | "bound0 (NEq a) = numbound0 a"
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   291
  | "bound0 (Dvd i a) = numbound0 a"
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   292
  | "bound0 (NDvd i a) = numbound0 a"
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parents: 25162
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   293
  | "bound0 (NOT p) = bound0 p"
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parents: 25162
diff changeset
   294
  | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
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parents: 25162
diff changeset
   295
  | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
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parents: 25162
diff changeset
   296
  | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
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parents: 25162
diff changeset
   297
  | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
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   298
  | "bound0 (E p) = False"
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   299
  | "bound0 (A p) = False"
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diff changeset
   300
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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   301
lemma bound0_I:
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   302
  assumes bp: "bound0 p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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diff changeset
   303
  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
51369
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wenzelm
parents: 51272
diff changeset
   304
  using bp numbound0_I [where b="b" and bs="bs" and b'="b'"]
41849
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nipkow
parents: 41839
diff changeset
   305
  by (induct p) auto
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parents: 25162
diff changeset
   306
49580bd58a21 some more primrec
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diff changeset
   307
primrec numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) where
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chaieb
parents:
diff changeset
   308
  "numsubst0 t (C c) = (C c)"
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   309
  | "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
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diff changeset
   310
  | "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))"
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haftmann
parents: 25162
diff changeset
   311
  | "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)"
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haftmann
parents: 25162
diff changeset
   312
  | "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
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haftmann
parents: 25162
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   313
  | "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   314
  | "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)"
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haftmann
parents: 25162
diff changeset
   315
  | "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"
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parents: 25162
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   316
  | "numsubst0 t (Floor a) = Floor (numsubst0 t a)"
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chaieb
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diff changeset
   317
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
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diff changeset
   318
lemma numsubst0_I:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   319
  shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
41849
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nipkow
parents: 41839
diff changeset
   320
  by (induct t) simp_all
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chaieb
parents:
diff changeset
   321
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   322
primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) where
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chaieb
parents:
diff changeset
   323
  "subst0 t T = T"
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   324
  | "subst0 t F = F"
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parents: 25162
diff changeset
   325
  | "subst0 t (Lt a) = Lt (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   326
  | "subst0 t (Le a) = Le (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   327
  | "subst0 t (Gt a) = Gt (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   328
  | "subst0 t (Ge a) = Ge (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   329
  | "subst0 t (Eq a) = Eq (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   330
  | "subst0 t (NEq a) = NEq (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   331
  | "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   332
  | "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)"
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haftmann
parents: 25162
diff changeset
   333
  | "subst0 t (NOT p) = NOT (subst0 t p)"
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haftmann
parents: 25162
diff changeset
   334
  | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   335
  | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
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haftmann
parents: 25162
diff changeset
   336
  | "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)"
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haftmann
parents: 25162
diff changeset
   337
  | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
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chaieb
parents:
diff changeset
   338
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   339
lemma subst0_I: assumes qfp: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   340
  shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   341
  using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
41849
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nipkow
parents: 41839
diff changeset
   342
  by (induct p) simp_all
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chaieb
parents:
diff changeset
   343
41839
421a795cee05 recdef -> fun(ction)
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diff changeset
   344
fun decrnum:: "num \<Rightarrow> num" where
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chaieb
parents:
diff changeset
   345
  "decrnum (Bound n) = Bound (n - 1)"
41839
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parents: 41836
diff changeset
   346
| "decrnum (Neg a) = Neg (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   347
| "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   348
| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   349
| "decrnum (Mul c a) = Mul c (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   350
| "decrnum (Floor a) = Floor (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   351
| "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   352
| "decrnum (CF c a b) = CF c (decrnum a) (decrnum b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   353
| "decrnum a = a"
421a795cee05 recdef -> fun(ction)
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diff changeset
   354
421a795cee05 recdef -> fun(ction)
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parents: 41836
diff changeset
   355
fun decr :: "fm \<Rightarrow> fm" where
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chaieb
parents:
diff changeset
   356
  "decr (Lt a) = Lt (decrnum a)"
41839
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krauss
parents: 41836
diff changeset
   357
| "decr (Le a) = Le (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   358
| "decr (Gt a) = Gt (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   359
| "decr (Ge a) = Ge (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   360
| "decr (Eq a) = Eq (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   361
| "decr (NEq a) = NEq (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   362
| "decr (Dvd i a) = Dvd i (decrnum a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   363
| "decr (NDvd i a) = NDvd i (decrnum a)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   364
| "decr (NOT p) = NOT (decr p)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   365
| "decr (And p q) = And (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   366
| "decr (Or p q) = Or (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   367
| "decr (Imp p q) = Imp (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   368
| "decr (Iff p q) = Iff (decr p) (decr q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   369
| "decr p = p"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   370
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   371
lemma decrnum: assumes nb: "numbound0 t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   372
  shows "Inum (x#bs) t = Inum bs (decrnum t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   373
  using nb by (induct t rule: decrnum.induct) simp_all
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   374
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   375
lemma decr: assumes nb: "bound0 p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   376
  shows "Ifm (x#bs) p = Ifm bs (decr p)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   377
  using nb by (induct p rule: decr.induct) (simp_all add: decrnum)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   378
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   379
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   380
  by (induct p) simp_all
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   381
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   382
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   383
  "isatom T = True"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   384
| "isatom F = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   385
| "isatom (Lt a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   386
| "isatom (Le a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   387
| "isatom (Gt a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   388
| "isatom (Ge a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   389
| "isatom (Eq a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   390
| "isatom (NEq a) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   391
| "isatom (Dvd i b) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   392
| "isatom (NDvd i b) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   393
| "isatom p = False"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   394
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   395
lemma numsubst0_numbound0:
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   396
  assumes nb: "numbound0 t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   397
  shows "numbound0 (numsubst0 t a)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   398
  using nb by (induct a) auto
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   399
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   400
lemma subst0_bound0:
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   401
  assumes qf: "qfree p" and nb: "numbound0 t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   402
  shows "bound0 (subst0 t p)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   403
  using qf numsubst0_numbound0[OF nb] by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   404
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   405
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   406
  by (induct p) simp_all
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   407
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   408
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   409
definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   410
  "djf f p q = (if q=T then T else if q=F then f p else
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   411
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q))"
25765
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   412
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   413
definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
49580bd58a21 some more primrec
haftmann
parents: 25162
diff changeset
   414
  "evaldjf f ps = foldr (djf f) ps F"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   415
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   416
lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   417
  by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   418
  (cases "f p", simp_all add: Let_def djf_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   419
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   420
lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   421
  by (induct ps) (simp_all add: evaldjf_def djf_Or)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   422
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   423
lemma evaldjf_bound0:
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   424
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   425
  shows "bound0 (evaldjf f xs)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   426
  using nb
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   427
  apply (induct xs)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   428
  apply (auto simp add: evaldjf_def djf_def Let_def)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   429
  apply (case_tac "f a")
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   430
  apply auto
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   431
  done
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   432
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   433
lemma evaldjf_qf:
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   434
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   435
  shows "qfree (evaldjf f xs)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   436
  using nb
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   437
  apply (induct xs)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   438
  apply (auto simp add: evaldjf_def djf_def Let_def)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   439
  apply (case_tac "f a")
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   440
  apply auto
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   441
  done
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   442
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   443
fun disjuncts :: "fm \<Rightarrow> fm list" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   444
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   445
| "disjuncts F = []"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   446
| "disjuncts p = [p]"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   447
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   448
fun conjuncts :: "fm \<Rightarrow> fm list" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   449
  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   450
| "conjuncts T = []"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   451
| "conjuncts p = [p]"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   452
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   453
lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   454
  by (induct p rule: conjuncts.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   455
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   456
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   457
proof -
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   458
  assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   459
  hence "list_all qfree (disjuncts p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   460
    by (induct p rule: disjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   461
  thus ?thesis by (simp only: list_all_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   462
qed
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   463
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   464
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   465
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   466
  assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   467
  hence "list_all qfree (conjuncts p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   468
    by (induct p rule: conjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   469
  thus ?thesis by (simp only: list_all_iff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   470
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   471
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   472
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   473
  "DJ f p \<equiv> evaldjf f (disjuncts p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   474
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   475
lemma DJ: assumes fdj: "\<forall> p q. f (Or p q) = Or (f p) (f q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   476
  and fF: "f F = F"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   477
  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   478
proof -
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   479
  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   480
    by (simp add: DJ_def evaldjf_ex)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   481
  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   482
  finally show ?thesis .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   483
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   484
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   485
lemma DJ_qf: assumes
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   486
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   487
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   488
proof(clarify)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   489
  fix  p assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   490
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   491
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   492
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   493
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   494
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   495
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   496
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   497
lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   498
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   499
proof(clarify)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   500
  fix p::fm and bs
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   501
  assume qf: "qfree p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   502
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   503
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   504
  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   505
    by (simp add: DJ_def evaldjf_ex)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   506
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   507
  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   508
  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   509
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   510
  (* Simplification *)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   511
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   512
  (* Algebraic simplifications for nums *)
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   513
fun bnds:: "num \<Rightarrow> nat list" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   514
  "bnds (Bound n) = [n]"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   515
| "bnds (CN n c a) = n#(bnds a)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   516
| "bnds (Neg a) = bnds a"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   517
| "bnds (Add a b) = (bnds a)@(bnds b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   518
| "bnds (Sub a b) = (bnds a)@(bnds b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   519
| "bnds (Mul i a) = bnds a"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   520
| "bnds (Floor a) = bnds a"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   521
| "bnds (CF c a b) = (bnds a)@(bnds b)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   522
| "bnds a = []"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   523
fun lex_ns:: "nat list \<Rightarrow> nat list \<Rightarrow> bool" where
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   524
  "lex_ns [] ms = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   525
| "lex_ns ns [] = False"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   526
| "lex_ns (n#ns) (m#ms) = (n<m \<or> ((n = m) \<and> lex_ns ns ms)) "
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   527
definition lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool" where
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   528
  "lex_bnd t s \<equiv> lex_ns (bnds t) (bnds s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   529
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   530
fun maxcoeff:: "num \<Rightarrow> int" where
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   531
  "maxcoeff (C i) = \<bar>i\<bar>"
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   532
| "maxcoeff (CN n c t) = max \<bar>c\<bar> (maxcoeff t)"
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   533
| "maxcoeff (CF c t s) = max \<bar>c\<bar> (maxcoeff s)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   534
| "maxcoeff t = 1"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   535
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   536
lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   537
  by (induct t rule: maxcoeff.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   538
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   539
fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" where
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   540
  "numgcdh (C i) = (\<lambda>g. gcd i g)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   541
| "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   542
| "numgcdh (CF c s t) = (\<lambda>g. gcd c (numgcdh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   543
| "numgcdh t = (\<lambda>g. 1)"
23858
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   544
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   545
definition numgcd :: "num \<Rightarrow> int"
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   546
  where "numgcd t = numgcdh t (maxcoeff t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   547
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   548
fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   549
  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   550
| "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   551
| "reducecoeffh (CF c s t) = (\<lambda> g. CF (c div g)  s (reducecoeffh t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   552
| "reducecoeffh t = (\<lambda>g. t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   553
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   554
definition reducecoeff :: "num \<Rightarrow> num"
23858
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   555
where
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   556
  "reducecoeff t =
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   557
    (let g = numgcd t in
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   558
     if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   559
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   560
fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   561
  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   562
| "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   563
| "dvdnumcoeff (CF c s t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   564
| "dvdnumcoeff t = (\<lambda>g. False)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   565
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   566
lemma dvdnumcoeff_trans:
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   567
  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   568
  shows "dvdnumcoeff t g"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   569
  using dgt' gdg
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   570
  by (induct t rule: dvdnumcoeff.induct) (simp_all add: gdg dvd_trans[OF gdg])
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29823
diff changeset
   571
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29823
diff changeset
   572
declare dvd_trans [trans add]
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   573
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   574
lemma numgcd0:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   575
  assumes g0: "numgcd t = 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   576
  shows "Inum bs t = 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   577
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   578
  have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   579
    by (induct t rule: numgcdh.induct, auto)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   580
  thus ?thesis using g0[simplified numgcd_def] by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   581
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   582
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   583
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   584
  using gp by (induct t rule: numgcdh.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   585
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   586
lemma numgcd_pos: "numgcd t \<ge>0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   587
  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   588
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   589
lemma reducecoeffh:
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   590
  assumes gt: "dvdnumcoeff t g" and gp: "g > 0"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   591
  shows "real_of_int g *(Inum bs (reducecoeffh t g)) = Inum bs t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   592
  using gt
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   593
proof(induct t rule: reducecoeffh.induct)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   594
  case (1 i) hence gd: "g dvd i" by simp
46670
e9aa6d151329 removing unnecessary assumptions in RealDef;
bulwahn
parents: 46130
diff changeset
   595
  from assms 1 show ?case by (simp add: real_of_int_div[OF gd])
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   596
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   597
  case (2 n c t)  hence gd: "g dvd c" by simp
46670
e9aa6d151329 removing unnecessary assumptions in RealDef;
bulwahn
parents: 46130
diff changeset
   598
  from assms 2 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   599
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   600
  case (3 c s t)  hence gd: "g dvd c" by simp
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   601
  from assms 3 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   602
qed (auto simp add: numgcd_def gp)
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   603
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   604
fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   605
  "ismaxcoeff (C i) = (\<lambda> x. \<bar>i\<bar> \<le> x)"
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   606
| "ismaxcoeff (CN n c t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> (ismaxcoeff t x))"
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   607
| "ismaxcoeff (CF c s t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> (ismaxcoeff t x))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   608
| "ismaxcoeff t = (\<lambda>x. True)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   609
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   610
lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   611
  by (induct t rule: ismaxcoeff.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   612
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   613
lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   614
proof (induct t rule: maxcoeff.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   615
  case (2 n c t)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   616
  hence H:"ismaxcoeff t (maxcoeff t)" .
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   617
  have thh: "maxcoeff t \<le> max \<bar>c\<bar> (maxcoeff t)" by simp
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   618
  from ismaxcoeff_mono[OF H thh] show ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   619
next
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   620
  case (3 c t s)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   621
  hence H1:"ismaxcoeff s (maxcoeff s)" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   622
  have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def)
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   623
  from ismaxcoeff_mono[OF H1 thh1] show ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   624
qed simp_all
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   625
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   626
lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((\<bar>i\<bar> > 1 \<and> \<bar>j\<bar> > 1) \<or> (\<bar>i\<bar> = 0 \<and> \<bar>j\<bar> > 1) \<or> (\<bar>i\<bar> > 1 \<and> \<bar>j\<bar> = 0))"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   627
  apply (unfold gcd_int_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   628
  apply (cases "i = 0", simp_all)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   629
  apply (cases "j = 0", simp_all)
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   630
  apply (cases "\<bar>i\<bar> = 1", simp_all)
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   631
  apply (cases "\<bar>j\<bar> = 1", simp_all)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   632
  apply auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   633
  done
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   634
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   635
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   636
  by (induct t rule: numgcdh.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   637
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   638
lemma dvdnumcoeff_aux:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   639
  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   640
  shows "dvdnumcoeff t (numgcdh t m)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   641
using assms
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   642
proof(induct t rule: numgcdh.induct)
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   643
  case (2 n c t)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   644
  let ?g = "numgcdh t m"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   645
  from 2 have th:"gcd c ?g > 1" by simp
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27456
diff changeset
   646
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   647
  have "(\<bar>c\<bar> > 1 \<and> ?g > 1) \<or> (\<bar>c\<bar> = 0 \<and> ?g > 1) \<or> (\<bar>c\<bar> > 1 \<and> ?g = 0)" by simp
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   648
  moreover {assume "\<bar>c\<bar> > 1" and gp: "?g > 1" with 2
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   649
    have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   650
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   651
    from dvdnumcoeff_trans[OF th' th] have ?case by simp }
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   652
  moreover {assume "\<bar>c\<bar> = 0 \<and> ?g > 1"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   653
    with 2 have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   654
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   655
    from dvdnumcoeff_trans[OF th' th] have ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   656
    hence ?case by simp }
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   657
  moreover {assume "\<bar>c\<bar> > 1" and g0:"?g = 0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   658
    from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   659
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   660
next
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   661
  case (3 c s t)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   662
  let ?g = "numgcdh t m"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   663
  from 3 have th:"gcd c ?g > 1" by simp
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27456
diff changeset
   664
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   665
  have "(\<bar>c\<bar> > 1 \<and> ?g > 1) \<or> (\<bar>c\<bar> = 0 \<and> ?g > 1) \<or> (\<bar>c\<bar> > 1 \<and> ?g = 0)" by simp
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   666
  moreover {assume "\<bar>c\<bar> > 1" and gp: "?g > 1" with 3
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   667
    have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   668
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   669
    from dvdnumcoeff_trans[OF th' th] have ?case by simp }
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   670
  moreover {assume "\<bar>c\<bar> = 0 \<and> ?g > 1"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   671
    with 3 have th: "dvdnumcoeff t ?g" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   672
    have th': "gcd c ?g dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   673
    from dvdnumcoeff_trans[OF th' th] have ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   674
    hence ?case by simp }
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   675
  moreover {assume "\<bar>c\<bar> > 1" and g0:"?g = 0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   676
    from numgcdh0[OF g0] have "m=0". with 3 g0 have ?case by simp }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   677
  ultimately show ?case by blast
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   678
qed auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   679
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   680
lemma dvdnumcoeff_aux2:
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   681
  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   682
  using assms
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   683
proof (simp add: numgcd_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   684
  let ?mc = "maxcoeff t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   685
  let ?g = "numgcdh t ?mc"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   686
  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   687
  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   688
  assume H: "numgcdh t ?mc > 1"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   689
  from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" .
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   690
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   691
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   692
lemma reducecoeff: "real_of_int (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   693
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   694
  let ?g = "numgcd t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   695
  have "?g \<ge> 0"  by (simp add: numgcd_pos)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   696
  hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   697
  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)}
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   698
  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)}
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   699
  moreover { assume g1:"?g > 1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   700
    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   701
    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   702
      by (simp add: reducecoeff_def Let_def)}
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   703
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   704
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   705
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   706
lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   707
  by (induct t rule: reducecoeffh.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   708
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   709
lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   710
  using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   711
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   712
consts numadd:: "num \<times> num \<Rightarrow> num"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   713
recdef numadd "measure (\<lambda> (t,s). size t + size s)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   714
  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   715
  (if n1=n2 then
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   716
  (let c = c1 + c2
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   717
  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   718
  else if n1 \<le> n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   719
  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   720
  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   721
  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   722
  "numadd (CF c1 t1 r1,CF c2 t2 r2) =
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   723
   (if t1 = t2 then
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   724
    (let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   725
   else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2))
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   726
   else CF c2 t2 (numadd(CF c1 t1 r1,r2)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   727
  "numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   728
  "numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   729
  "numadd (C b1, C b2) = C (b1+b2)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   730
  "numadd (a,b) = Add a b"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   731
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   732
lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   733
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   734
 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29265
diff changeset
   735
  apply (case_tac "n1 = n2", simp_all add: algebra_simps)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 49069
diff changeset
   736
  apply (simp only: distrib_right[symmetric])
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   737
 apply simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   738
apply (case_tac "lex_bnd t1 t2", simp_all)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23464
diff changeset
   739
 apply (case_tac "c1+c2 = 0")
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   740
  apply (case_tac "t1 = t2")
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   741
   apply (simp_all add: algebra_simps distrib_right[symmetric] of_int_mult[symmetric] of_int_add[symmetric]del: of_int_mult of_int_add distrib_right)
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   742
  done
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   743
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   744
lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   745
  by (induct t s rule: numadd.induct) (auto simp add: Let_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   746
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   747
fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   748
  "nummul (C j) = (\<lambda> i. C (i*j))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   749
| "nummul (CN n c t) = (\<lambda> i. CN n (c*i) (nummul t i))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   750
| "nummul (CF c t s) = (\<lambda> i. CF (c*i) t (nummul s i))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   751
| "nummul (Mul c t) = (\<lambda> i. nummul t (i*c))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   752
| "nummul t = (\<lambda> i. Mul i t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   753
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   754
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   755
  by (induct t rule: nummul.induct) (auto simp add: algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   756
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   757
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   758
  by (induct t rule: nummul.induct) auto
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   759
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   760
definition numneg :: "num \<Rightarrow> num"
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   761
  where "numneg t \<equiv> nummul t (- 1)"
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   762
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   763
definition numsub :: "num \<Rightarrow> num \<Rightarrow> num"
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   764
  where "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   765
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   766
lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   767
  using numneg_def nummul by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   768
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   769
lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   770
  using numneg_def by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   771
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   772
lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   773
  using numsub_def by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   774
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   775
lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   776
  using numsub_def by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   777
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   778
lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   779
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   780
  have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor)
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   781
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   782
  have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   783
  also have "\<dots>" by (simp add: isint_add cti si)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   784
  finally show ?thesis .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   785
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   786
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   787
fun split_int:: "num \<Rightarrow> num \<times> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   788
  "split_int (C c) = (C 0, C c)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   789
| "split_int (CN n c b) =
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   790
     (let (bv,bi) = split_int b
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   791
       in (CN n c bv, bi))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   792
| "split_int (CF c a b) =
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   793
     (let (bv,bi) = split_int b
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   794
       in (bv, CF c a bi))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   795
| "split_int a = (a,C 0)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   796
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   797
lemma split_int: "\<And>tv ti. split_int t = (tv,ti) \<Longrightarrow> (Inum bs (Add tv ti) = Inum bs t) \<and> isint ti bs"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   798
proof (induct t rule: split_int.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   799
  case (2 c n b tv ti)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   800
  let ?bv = "fst (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   801
  let ?bi = "snd (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   802
  have "split_int b = (?bv,?bi)" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   803
  with 2(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   804
  from 2(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def)
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   805
  from 2(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   806
next
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   807
  case (3 c a b tv ti)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   808
  let ?bv = "fst (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   809
  let ?bi = "snd (split_int b)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   810
  have "split_int b = (?bv,?bi)" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   811
  with 3(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   812
  from 3(2) have tibi: "ti = CF c a ?bi"
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   813
    by (simp add: Let_def split_def)
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   814
  from 3(2) b[symmetric] bii show ?case
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   815
    by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29265
diff changeset
   816
qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   817
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   818
lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) "
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   819
  by (induct t rule: split_int.induct) (auto simp add: Let_def split_def)
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   820
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   821
definition numfloor:: "num \<Rightarrow> num"
23858
5500610fe1e5 adapted to new code generator framework
haftmann
parents: 23477
diff changeset
   822
where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   823
  "numfloor t = (let (tv,ti) = split_int t in
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   824
  (case tv of C i \<Rightarrow> numadd (tv,ti)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   825
  | _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   826
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   827
lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)")
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   828
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   829
  let ?tv = "fst (split_int t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   830
  let ?ti = "snd (split_int t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   831
  have tvti:"split_int t = (?tv,?ti)" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   832
  {assume H: "\<forall> v. ?tv \<noteq> C v"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   833
    hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   834
      by (cases ?tv) (auto simp add: numfloor_def Let_def split_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   835
    from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   836
    hence "?N (Floor t) = real_of_int \<lfloor>?N (Add ?tv ?ti)\<rfloor>" by simp
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   837
    also have "\<dots> = real_of_int (\<lfloor>?N ?tv\<rfloor> + \<lfloor>?N ?ti\<rfloor>)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   838
      by (simp,subst tii[simplified isint_iff, symmetric]) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   839
    also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   840
    finally have ?thesis using th1 by simp}
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   841
  moreover {fix v assume H:"?tv = C v"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   842
    from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   843
    hence "?N (Floor t) = real_of_int \<lfloor>?N (Add ?tv ?ti)\<rfloor>" by simp
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61762
diff changeset
   844
    also have "\<dots> = real_of_int (\<lfloor>?N ?tv\<rfloor> + \<lfloor>?N ?ti\<rfloor>)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   845
      by (simp,subst tii[simplified isint_iff, symmetric]) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   846
    also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   847
    finally have ?thesis by (simp add: H numfloor_def Let_def split_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   848
  ultimately show ?thesis by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   849
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   850
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   851
lemma numfloor_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numfloor t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   852
  using split_int_nb[where t="t"]
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   853
  by (cases "fst (split_int t)") (auto simp add: numfloor_def Let_def split_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   854
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   855
function simpnum:: "num \<Rightarrow> num" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   856
  "simpnum (C j) = C j"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   857
| "simpnum (Bound n) = CN n 1 (C 0)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   858
| "simpnum (Neg t) = numneg (simpnum t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   859
| "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   860
| "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   861
| "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   862
| "simpnum (Floor t) = numfloor (simpnum t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   863
| "simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   864
| "simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   865
by pat_completeness auto
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   866
termination by (relation "measure num_size") auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   867
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   868
lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   869
  by (induct t rule: simpnum.induct) auto
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   870
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   871
lemma simpnum_numbound0[simp]: "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   872
  by (induct t rule: simpnum.induct) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   873
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   874
fun nozerocoeff:: "num \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   875
  "nozerocoeff (C c) = True"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   876
| "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   877
| "nozerocoeff (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   878
| "nozerocoeff (Mul c t) = (c\<noteq>0 \<and> nozerocoeff t)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
   879
| "nozerocoeff t = True"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   880
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   881
lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   882
  by (induct a b rule: numadd.induct) (auto simp add: Let_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   883
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   884
lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   885
  by (induct a rule: nummul.induct) (auto simp add: Let_def numadd_nz)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   886
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   887
lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   888
  by (simp add: numneg_def nummul_nz)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   889
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   890
lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   891
  by (simp add: numsub_def numneg_nz numadd_nz)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   892
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   893
lemma split_int_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   894
  by (induct t rule: split_int.induct) (auto simp add: Let_def split_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   895
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   896
lemma numfloor_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (numfloor t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   897
  by (simp add: numfloor_def Let_def split_def)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   898
    (cases "fst (split_int t)", simp_all add: split_int_nz numadd_nz)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   899
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   900
lemma simpnum_nz: "nozerocoeff (simpnum t)"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   901
  by (induct t rule: simpnum.induct)
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
   902
    (auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz numfloor_nz)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   903
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   904
lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   905
proof (induct t rule: maxcoeff.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   906
  case (2 n c t)
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   907
  hence cnz: "c \<noteq>0" and mx: "max \<bar>c\<bar> (maxcoeff t) = 0" by simp+
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   908
  have "max \<bar>c\<bar> (maxcoeff t) \<ge> \<bar>c\<bar>" by simp
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   909
  with cnz have "max \<bar>c\<bar> (maxcoeff t) > 0" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   910
  with 2 show ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   911
next
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   912
  case (3 c s t)
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   913
  hence cnz: "c \<noteq>0" and mx: "max \<bar>c\<bar> (maxcoeff t) = 0" by simp+
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   914
  have "max \<bar>c\<bar> (maxcoeff t) \<ge> \<bar>c\<bar>" by simp
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
   915
  with cnz have "max \<bar>c\<bar> (maxcoeff t) > 0" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   916
  with 3 show ?case by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   917
qed auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   918
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   919
lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   920
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   921
  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   922
  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   923
  from maxcoeff_nz[OF nz th] show ?thesis .
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   924
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   925
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
   926
definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   927
  "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   928
   (let t' = simpnum t ; g = numgcd t' in
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   929
      if g > 1 then (let g' = gcd n g in
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   930
        if g' = 1 then (t',n)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   931
        else (reducecoeffh t' g', n div g'))
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   932
      else (t',n))))"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   933
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   934
lemma simp_num_pair_ci:
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   935
  shows "((\<lambda> (t,n). Inum bs t / real_of_int n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real_of_int n) (t,n))"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   936
  (is "?lhs = ?rhs")
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   937
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   938
  let ?t' = "simpnum t"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   939
  let ?g = "numgcd ?t'"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   940
  let ?g' = "gcd n ?g"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   941
  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   942
  moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   943
  { assume nnz: "n \<noteq> 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   944
    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   945
    moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   946
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   947
      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31730
diff changeset
   948
      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   949
      hence "?g'= 1 \<or> ?g' > 1" by arith
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   950
      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   951
      moreover {assume g'1:"?g'>1"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   952
        from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   953
        let ?tt = "reducecoeffh ?t' ?g'"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   954
        let ?t = "Inum bs ?tt"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   955
        have gpdg: "?g' dvd ?g" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   956
        have gpdd: "?g' dvd n" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   957
        have gpdgp: "?g' dvd ?g'" by simp
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   958
        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   959
        have th2:"real_of_int ?g' * ?t = Inum bs ?t'" by simp
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   960
        from nnz g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')" by (simp add: simp_num_pair_def Let_def)
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   961
        also have "\<dots> = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))" by simp
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   962
        also have "\<dots> = (Inum bs ?t' / real_of_int n)"
46670
e9aa6d151329 removing unnecessary assumptions in RealDef;
bulwahn
parents: 46130
diff changeset
   963
          using real_of_int_div[OF gpdd] th2 gp0 by simp
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
   964
        finally have "?lhs = Inum bs t / real_of_int n" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   965
        then have ?thesis using nnz g1 g'1 by (simp add: simp_num_pair_def) }
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   966
      ultimately have ?thesis by blast }
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   967
    ultimately have ?thesis by blast }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   968
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   969
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   970
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   971
lemma simp_num_pair_l:
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   972
  assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   973
  shows "numbound0 t' \<and> n' >0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   974
proof-
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   975
  let ?t' = "simpnum t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   976
  let ?g = "numgcd ?t'"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   977
  let ?g' = "gcd n ?g"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   978
  { assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   979
  moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   980
  { assume nnz: "n \<noteq> 0"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   981
    {assume "\<not> ?g > 1" hence ?thesis using assms by (auto simp add: Let_def simp_num_pair_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   982
    moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   983
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
   984
      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31730
diff changeset
   985
      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   986
      hence "?g'= 1 \<or> ?g' > 1" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   987
      moreover {assume "?g'=1" hence ?thesis using assms g1 g0
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   988
          by (auto simp add: Let_def simp_num_pair_def) }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
   989
      moreover {assume g'1:"?g'>1"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   990
        have gpdg: "?g' dvd ?g" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   991
        have gpdd: "?g' dvd n" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   992
        have gpdgp: "?g' dvd ?g'" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   993
        from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
47142
d64fa2ca54b8 remove redundant lemmas
huffman
parents: 47108
diff changeset
   994
        from zdiv_mono1[OF g'n g'p, simplified div_self[OF gp0]]
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   995
        have "n div ?g' >0" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   996
        hence ?thesis using assms g1 g'1
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
   997
          by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)}
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
   998
      ultimately have ?thesis by blast }
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
   999
    ultimately have ?thesis by blast }
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1000
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1001
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1002
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1003
fun not:: "fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1004
  "not (NOT p) = p"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1005
| "not T = F"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1006
| "not F = T"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1007
| "not (Lt t) = Ge t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1008
| "not (Le t) = Gt t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1009
| "not (Gt t) = Le t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1010
| "not (Ge t) = Lt t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1011
| "not (Eq t) = NEq t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1012
| "not (NEq t) = Eq t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1013
| "not (Dvd i t) = NDvd i t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1014
| "not (NDvd i t) = Dvd i t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1015
| "not (And p q) = Or (not p) (not q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1016
| "not (Or p q) = And (not p) (not q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1017
| "not p = NOT p"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1018
lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1019
  by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1020
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1021
  by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1022
lemma not_nb[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1023
  by (induct p) auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1024
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1025
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1026
  "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
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1027
   if p = q then p else And p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1028
lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1029
  by (cases "p=F \<or> q=F", simp_all add: conj_def) (cases p, simp_all)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1030
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1031
lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1032
  using conj_def by auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1033
lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1034
  using conj_def by auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1035
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1036
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1037
  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1038
       else if p=q then p else Or p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1039
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1040
lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1041
  by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1042
lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1043
  using disj_def by auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1044
lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1045
  using disj_def by auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1046
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1047
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1048
  "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
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1049
    else Imp p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1050
lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1051
  by (cases "p=F \<or> q=T",simp_all add: imp_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1052
lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1053
  using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1054
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1055
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1056
  "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1057
       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
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1058
  Iff p q)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61610
diff changeset
  1059
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1060
lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61610
diff changeset
  1061
  by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp)  (cases "not p= q", auto simp add:not)
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61610
diff changeset
  1062
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1063
lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1064
  by (unfold iff_def,cases "p=q", auto)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1065
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1066
fun check_int:: "num \<Rightarrow> bool" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1067
  "check_int (C i) = True"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1068
| "check_int (Floor t) = True"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1069
| "check_int (Mul i t) = check_int t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1070
| "check_int (Add t s) = (check_int t \<and> check_int s)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1071
| "check_int (Neg t) = check_int t"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1072
| "check_int (CF c t s) = check_int s"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1073
| "check_int t = False"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1074
lemma check_int: "check_int t \<Longrightarrow> isint t bs"
51369
960b0ca9ae5d tuned proofs -- more structure, less warnings;
wenzelm
parents: 51272
diff changeset
  1075
  by (induct t) (auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF)
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1076
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1077
lemma rdvd_left1_int: "real_of_int \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1078
  by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1079
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1080
lemma rdvd_reduce:
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1081
  assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1082
  shows "real_of_int (d::int) rdvd real_of_int (c::int)*t = (real_of_int (d div g) rdvd real_of_int (c div g)*t)"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1083
proof
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1084
  assume d: "real_of_int d rdvd real_of_int c * t"
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1085
  from d rdvd_def obtain k where k_def: "real_of_int c * t = real_of_int d* real_of_int (k::int)" by auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1086
  from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1087
  from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1088
  from k_def kd_def kc_def have "real_of_int g * real_of_int kc * t = real_of_int g * real_of_int kd * real_of_int k" by simp
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1089
  hence "real_of_int kc * t = real_of_int kd * real_of_int k" using gp by simp
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1090
  hence th:"real_of_int kd rdvd real_of_int kc * t" using rdvd_def by blast
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1091
  from kd_def gp have th':"kd = d div g" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1092
  from kc_def gp have "kc = c div g" by simp
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1093
  with th th' show "real_of_int (d div g) rdvd real_of_int (c div g) * t" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1094
next
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1095
  assume d: "real_of_int (d div g) rdvd real_of_int (c div g) * t"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1096
  from gp have gnz: "g \<noteq> 0" by simp
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1097
  thus "real_of_int d rdvd real_of_int c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real_of_int (c div g) * t"] real_of_int_div[OF gd] real_of_int_div[OF gc] by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1098
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1099
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35028
diff changeset
  1100
definition simpdvd :: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" where
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1101
  "simpdvd d t \<equiv>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1102
   (let g = numgcd t in
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1103
      if g > 1 then (let g' = gcd d g in
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1104
        if g' = 1 then (d, t)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1105
        else (d div g',reducecoeffh t g'))
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1106
      else (d, t))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1107
lemma simpdvd:
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1108
  assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1109
  shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1110
proof-
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1111
  let ?g = "numgcd t"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1112
  let ?g' = "gcd d ?g"
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1113
  {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1114
  moreover
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1115
  {assume g1:"?g>1" hence g0: "?g > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1116
    from g1 dnz have gp0: "?g' \<noteq> 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31730
diff changeset
  1117
    hence g'p: "?g' > 0" using gcd_ge_0_int[where x="d" and y="numgcd t"] by arith
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1118
    hence "?g'= 1 \<or> ?g' > 1" by arith
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1119
    moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1120
    moreover {assume g'1:"?g'>1"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1121
      from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" ..
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1122
      let ?tt = "reducecoeffh t ?g'"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1123
      let ?t = "Inum bs ?tt"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1124
      have gpdg: "?g' dvd ?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30649
diff changeset
  1125
      have gpdd: "?g' dvd d" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1126
      have gpdgp: "?g' dvd ?g'" by simp
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1127
      from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1128
      have th2:"real_of_int ?g' * ?t = Inum bs t" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1129
      from assms g1 g0 g'1
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41464
diff changeset
  1130
      have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1131
        by (simp add: simpdvd_def Let_def)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1132
      also have "\<dots> = (real_of_int d rdvd (Inum bs t))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1133
        using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified div_self[OF gp0]]
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31952
diff changeset
  1134
          th2[symmetric] by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1135
      finally have ?thesis by simp  }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1136
    ultimately have ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1137
  }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1138
  ultimately show ?thesis by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1139
qed
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1140
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1141
function (sequential) simpfm :: "fm \<Rightarrow> fm" where
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1142
  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1143
| "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1144
| "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1145
| "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1146
| "simpfm (NOT p) = not (simpfm p)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1147
| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1148
  | _ \<Rightarrow> Lt (reducecoeff a'))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1149
| "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1150
| "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1151
| "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1152
| "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1153
| "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq (reducecoeff a'))"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1154
| "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
  1155
             else if (\<bar>i\<bar> = 1) \<and> check_int a then T
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1156
             else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61652
diff changeset
  1157
| "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a)
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61942
diff changeset
  1158
             else if (\<bar>i\<bar> = 1) \<and> check_int a then F
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1159
             else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))"
41839
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1160
| "simpfm p = p"
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1161
by pat_completeness auto
421a795cee05 recdef -> fun(ction)
krauss
parents: 41836
diff changeset
  1162
termination by (relation "measure fmsize") auto
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1163
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1164
lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1165
proof(induct p rule: simpfm.induct)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1166
  case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1167
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1168
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1169
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1170
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1171
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1172
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1173
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1174
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1175
    hence gp: "real_of_int ?g > 0" by simp
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1176
    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1177
    with sa have "Inum bs a < 0 = (real_of_int ?g * ?r < real_of_int ?g * 0)" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1178
    also have "\<dots> = (?r < 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1179
      by (simp only: mult_less_cancel_left) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1180
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1181
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1182
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1183
  case (7 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1184
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1185
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1186
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1187
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1188
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1189
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1190
    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1191
    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1192
    hence gp: "real_of_int ?g > 0" by simp
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1193
    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61424
diff changeset
  1194
    with sa have "Inum bs a \<le> 0 = (real_of_int ?g * ?r \<le> real_of_int ?g * 0)" by simp
23264
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1195
    also have "\<dots> = (?r \<le> 0)" using gp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1196
      by (simp only: mult_le_cancel_left) simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1197
    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1198
  ultimately show ?case by blast
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1199
next
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1200
  case (8 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1201
  {fix v assume "?sa = C v" hence ?case using sa by simp }
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1202
  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1203
    let ?g = "numgcd ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1204
    let ?rsa = "reducecoeff ?sa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1205
    let ?r = "Inum bs ?rsa"
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1206
    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
324622260d29 Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff changeset
  1207
    {assume