author | huffman |
Fri, 09 Sep 2005 19:34:22 +0200 | |
changeset 17318 | bc1c75855f3d |
parent 17298 | ad73fb6144cf |
child 17429 | e8d6ed3aacfe |
permissions | -rw-r--r-- |
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(* Title: HOL/HyperArith.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1999 University of Cambridge |
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*) |
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||
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header{*Binary arithmetic and Simplification for the Hyperreals*} |
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||
15131 | 9 |
theory HyperArith |
15140 | 10 |
imports HyperDef |
16417 | 11 |
uses ("hypreal_arith.ML") |
15131 | 12 |
begin |
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13 |
|
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subsection{*Numerals and Arithmetic*} |
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|
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use "hypreal_arith.ML" |
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|
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setup hypreal_arith_setup |
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||
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Conversion of HyperNat to Isar format and its declaration as a semiring
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subsection{*Absolute Value Function for the Hyperreals*} |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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|
15003 | 22 |
lemma hrabs_add_less: |
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"[| abs x < r; abs y < s |] ==> abs(x+y) < r + (s::hypreal)" |
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24 |
by (simp add: abs_if split: split_if_asm) |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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25 |
|
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text{*used once in NSA*} |
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27 |
lemma hrabs_less_gt_zero: "abs x < r ==> (0::hypreal) < r" |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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by (blast intro!: order_le_less_trans abs_ge_zero) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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30 |
lemma hrabs_disj: "abs x = (x::hypreal) | abs x = -x" |
15003 | 31 |
by (simp add: abs_if) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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32 |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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(* Needed in Geom.ML *) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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lemma hrabs_add_lemma_disj: "(y::hypreal) + - x + (y + - z) = abs (x + - z) ==> y = z | x = y" |
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by (simp add: abs_if split add: split_if_asm) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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|
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lemma hypreal_of_real_hrabs: |
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"abs (hypreal_of_real r) = hypreal_of_real (abs r)" |
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by (rule star_of_abs [symmetric]) |
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|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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|
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subsection{*Embedding the Naturals into the Hyperreals*} |
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|
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constdefs |
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hypreal_of_nat :: "nat => hypreal" |
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"hypreal_of_nat m == of_nat m" |
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|
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lemma SNat_eq: "Nats = {n. \<exists>N. n = hypreal_of_nat N}" |
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by (force simp add: hypreal_of_nat_def Nats_def) |
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|
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lemma hypreal_of_nat_add [simp]: |
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"hypreal_of_nat (m + n) = hypreal_of_nat m + hypreal_of_nat n" |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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by (simp add: hypreal_of_nat_def) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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54 |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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lemma hypreal_of_nat_mult: "hypreal_of_nat (m * n) = hypreal_of_nat m * hypreal_of_nat n" |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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56 |
by (simp add: hypreal_of_nat_def) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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declare hypreal_of_nat_mult [simp] |
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|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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lemma hypreal_of_nat_less_iff: |
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"(n < m) = (hypreal_of_nat n < hypreal_of_nat m)" |
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apply (simp add: hypreal_of_nat_def) |
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done |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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63 |
declare hypreal_of_nat_less_iff [symmetric, simp] |
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64 |
|
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(*------------------------------------------------------------*) |
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(* naturals embedded in hyperreals *) |
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(* is a hyperreal c.f. NS extension *) |
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(*------------------------------------------------------------*) |
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69 |
|
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lemma hypreal_of_nat_eq: |
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"hypreal_of_nat (n::nat) = hypreal_of_real (real n)" |
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72 |
apply (induct n) |
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apply (simp_all add: hypreal_of_nat_def real_of_nat_def) |
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parents:
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74 |
done |
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parents:
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75 |
|
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paulson
parents:
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lemma hypreal_of_nat: |
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"hypreal_of_nat m = star_n (%n. real m)" |
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starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
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78 |
apply (fold star_of_def) |
17298 | 79 |
apply (induct m) |
17318
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huffman
parents:
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80 |
apply (simp_all add: hypreal_of_nat_def real_of_nat_def star_n_add) |
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paulson
parents:
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81 |
done |
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parents:
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82 |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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lemma hypreal_of_nat_Suc: |
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84 |
"hypreal_of_nat (Suc n) = hypreal_of_nat n + (1::hypreal)" |
14378
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generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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changeset
|
85 |
by (simp add: hypreal_of_nat_def) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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parents:
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86 |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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87 |
(*"neg" is used in rewrite rules for binary comparisons*) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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parents:
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88 |
lemma hypreal_of_nat_number_of [simp]: |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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"hypreal_of_nat (number_of v :: nat) = |
14378
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paulson
parents:
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changeset
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90 |
(if neg (number_of v :: int) then 0 |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
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91 |
else (number_of v :: hypreal))" |
14378
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generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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changeset
|
92 |
by (simp add: hypreal_of_nat_eq) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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|
93 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
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parents:
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94 |
lemma hypreal_of_nat_zero [simp]: "hypreal_of_nat 0 = 0" |
14378
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paulson
parents:
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changeset
|
95 |
by (simp add: hypreal_of_nat_def) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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96 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
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parents:
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|
97 |
lemma hypreal_of_nat_one [simp]: "hypreal_of_nat 1 = 1" |
14378
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paulson
parents:
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changeset
|
98 |
by (simp add: hypreal_of_nat_def) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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parents:
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|
99 |
|
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paulson
parents:
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100 |
lemma hypreal_of_nat_le_iff [simp]: |
69c4d5997669
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paulson
parents:
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|
101 |
"(hypreal_of_nat n \<le> hypreal_of_nat m) = (n \<le> m)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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changeset
|
102 |
by (simp add: hypreal_of_nat_def) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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parents:
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103 |
|
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paulson
parents:
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104 |
lemma hypreal_of_nat_ge_zero [simp]: "0 \<le> hypreal_of_nat n" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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105 |
by (simp add: hypreal_of_nat_def) |
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parents:
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106 |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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107 |
|
14309 | 108 |
(* |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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109 |
FIXME: we should declare this, as for type int, but many proofs would break. |
14309 | 110 |
It replaces x+-y by x-y. |
111 |
Addsimps [symmetric hypreal_diff_def] |
|
112 |
*) |
|
113 |
||
14369 | 114 |
ML |
115 |
{* |
|
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116 |
val hypreal_of_nat_def = thm"hypreal_of_nat_def"; |
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117 |
|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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118 |
val hrabs_add_less = thm "hrabs_add_less"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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119 |
val hrabs_disj = thm "hrabs_disj"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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120 |
val hrabs_add_lemma_disj = thm "hrabs_add_lemma_disj"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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121 |
val hypreal_of_real_hrabs = thm "hypreal_of_real_hrabs"; |
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paulson
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122 |
val hypreal_of_nat_add = thm "hypreal_of_nat_add"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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123 |
val hypreal_of_nat_mult = thm "hypreal_of_nat_mult"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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124 |
val hypreal_of_nat_less_iff = thm "hypreal_of_nat_less_iff"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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125 |
val hypreal_of_nat_Suc = thm "hypreal_of_nat_Suc"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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126 |
val hypreal_of_nat_number_of = thm "hypreal_of_nat_number_of"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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127 |
val hypreal_of_nat_zero = thm "hypreal_of_nat_zero"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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|
128 |
val hypreal_of_nat_one = thm "hypreal_of_nat_one"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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129 |
val hypreal_of_nat_le_iff = thm"hypreal_of_nat_le_iff"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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130 |
val hypreal_of_nat_ge_zero = thm"hypreal_of_nat_ge_zero"; |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
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131 |
val hypreal_of_nat = thm"hypreal_of_nat"; |
14369 | 132 |
*} |
133 |
||
10751 | 134 |
end |