author | haftmann |
Tue, 11 Dec 2007 10:23:08 +0100 | |
changeset 25601 | 24567e50ebcc |
parent 25162 | ad4d5365d9d8 |
child 27105 | 5f139027c365 |
permissions | -rw-r--r-- |
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(* Title : HyperNat.thy |
2 |
Author : Jacques D. Fleuriot |
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3 |
Copyright : 1998 University of Cambridge |
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Converted to Isar and polished by lcp |
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6 |
*) |
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|
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header{*Hypernatural numbers*} |
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9 |
|
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theory HyperNat |
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imports StarDef |
15131 | 12 |
begin |
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|
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types hypnat = "nat star" |
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|
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abbreviation |
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hypnat_of_nat :: "nat => nat star" where |
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"hypnat_of_nat == star_of" |
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|
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redefine hSuc as *f* Suc, and move to HyperNat.thy
huffman
parents:
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definition |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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hSuc :: "hypnat => hypnat" where |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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hSuc_def [transfer_unfold]: "hSuc = *f* Suc" |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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23 |
|
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subsection{*Properties Transferred from Naturals*} |
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25 |
|
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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26 |
lemma hSuc_not_zero [iff]: "\<And>m. hSuc m \<noteq> 0" |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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27 |
by transfer (rule Suc_not_Zero) |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
huffman
parents:
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28 |
|
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redefine hSuc as *f* Suc, and move to HyperNat.thy
huffman
parents:
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diff
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lemma zero_not_hSuc [iff]: "\<And>m. 0 \<noteq> hSuc m" |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
huffman
parents:
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30 |
by transfer (rule Zero_not_Suc) |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
huffman
parents:
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31 |
|
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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32 |
lemma hSuc_hSuc_eq [iff]: "\<And>m n. (hSuc m = hSuc n) = (m = n)" |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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parents:
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33 |
by transfer (rule Suc_Suc_eq) |
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redefine hSuc as *f* Suc, and move to HyperNat.thy
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lemma zero_less_hSuc [iff]: "\<And>n. 0 < hSuc n" |
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36 |
by transfer (rule zero_less_Suc) |
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37 |
|
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lemma hypnat_minus_zero [simp]: "!!z. z - z = (0::hypnat)" |
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by transfer (rule diff_self_eq_0) |
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|
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lemma hypnat_diff_0_eq_0 [simp]: "!!n. (0::hypnat) - n = 0" |
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by transfer (rule diff_0_eq_0) |
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43 |
|
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lemma hypnat_add_is_0 [iff]: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)" |
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by transfer (rule add_is_0) |
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|
17299 | 47 |
lemma hypnat_diff_diff_left: "!!i j k. (i::hypnat) - j - k = i - (j+k)" |
48 |
by transfer (rule diff_diff_left) |
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49 |
|
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lemma hypnat_diff_commute: "!!i j k. (i::hypnat) - j - k = i-k-j" |
51 |
by transfer (rule diff_commute) |
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52 |
|
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lemma hypnat_diff_add_inverse [simp]: "!!m n. ((n::hypnat) + m) - n = m" |
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by transfer (rule diff_add_inverse) |
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55 |
|
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lemma hypnat_diff_add_inverse2 [simp]: "!!m n. ((m::hypnat) + n) - n = m" |
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by transfer (rule diff_add_inverse2) |
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58 |
|
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59 |
lemma hypnat_diff_cancel [simp]: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n" |
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by transfer (rule diff_cancel) |
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61 |
|
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62 |
lemma hypnat_diff_cancel2 [simp]: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n" |
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by transfer (rule diff_cancel2) |
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64 |
|
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65 |
lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)" |
17299 | 66 |
by transfer (rule diff_add_0) |
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67 |
|
17299 | 68 |
lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - n) * k = (m * k) - (n * k)" |
69 |
by transfer (rule diff_mult_distrib) |
|
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70 |
|
17299 | 71 |
lemma hypnat_diff_mult_distrib2: "!!k m n. (k::hypnat) * (m - n) = (k * m) - (k * n)" |
72 |
by transfer (rule diff_mult_distrib2) |
|
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73 |
|
17299 | 74 |
lemma hypnat_le_zero_cancel [iff]: "!!n. (n \<le> (0::hypnat)) = (n = 0)" |
75 |
by transfer (rule le_0_eq) |
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76 |
|
17299 | 77 |
lemma hypnat_mult_is_0 [simp]: "!!m n. (m*n = (0::hypnat)) = (m=0 | n=0)" |
78 |
by transfer (rule mult_is_0) |
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79 |
|
17299 | 80 |
lemma hypnat_diff_is_0_eq [simp]: "!!m n. ((m::hypnat) - n = 0) = (m \<le> n)" |
81 |
by transfer (rule diff_is_0_eq) |
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|
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lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)" |
84 |
by transfer (rule not_less0) |
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|
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Conversion of HyperNat to Isar format and its declaration as a semiring
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86 |
lemma hypnat_less_one [iff]: |
17299 | 87 |
"!!n. (n < (1::hypnat)) = (n=0)" |
88 |
by transfer (rule less_one) |
|
89 |
||
90 |
lemma hypnat_add_diff_inverse: "!!m n. ~ m<n ==> n+(m-n) = (m::hypnat)" |
|
91 |
by transfer (rule add_diff_inverse) |
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92 |
|
17299 | 93 |
lemma hypnat_le_add_diff_inverse [simp]: "!!m n. n \<le> m ==> n+(m-n) = (m::hypnat)" |
94 |
by transfer (rule le_add_diff_inverse) |
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95 |
|
17299 | 96 |
lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n\<le>m ==> (m-n)+n = (m::hypnat)" |
97 |
by transfer (rule le_add_diff_inverse2) |
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98 |
|
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99 |
declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le] |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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100 |
|
17299 | 101 |
lemma hypnat_le0 [iff]: "!!n. (0::hypnat) \<le> n" |
102 |
by transfer (rule le0) |
|
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|
103 |
|
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104 |
lemma hypnat_le_add1 [simp]: "!!x n. (x::hypnat) \<le> x + n" |
5a103b43da5a
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|
105 |
by transfer (rule le_add1) |
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parents:
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|
106 |
|
17299 | 107 |
lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) \<le> n + x" |
108 |
by transfer (rule le_add2) |
|
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paulson
parents:
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diff
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|
109 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
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|
110 |
lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
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|
111 |
by (insert add_strict_left_mono [OF zero_less_one], auto) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
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|
112 |
|
17433 | 113 |
lemma hypnat_neq0_conv [iff]: "!!n. (n \<noteq> 0) = (0 < (n::hypnat))" |
114 |
by transfer (rule neq0_conv) |
|
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diff
changeset
|
115 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
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|
116 |
lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
117 |
by (auto simp add: linorder_not_less [symmetric]) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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diff
changeset
|
118 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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|
119 |
lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::hypnat))" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
120 |
apply safe |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
121 |
apply (rule_tac x = "n - (1::hypnat) " in exI) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
122 |
apply (simp add: hypnat_gt_zero_iff) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
123 |
apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
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|
124 |
done |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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diff
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|
125 |
|
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|
126 |
lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))" |
69c4d5997669
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paulson
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|
127 |
by (simp add: linorder_not_le [symmetric] add_commute [of x]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
128 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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|
129 |
lemma hypnat_diff_split: |
69c4d5997669
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paulson
parents:
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|
130 |
"P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))" |
69c4d5997669
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paulson
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|
131 |
-- {* elimination of @{text -} on @{text hypnat} *} |
69c4d5997669
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paulson
parents:
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|
132 |
proof (cases "a<b" rule: case_split) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
133 |
case True |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
134 |
thus ?thesis |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
135 |
by (auto simp add: hypnat_add_self_not_less order_less_imp_le |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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diff
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|
136 |
hypnat_diff_is_0_eq [THEN iffD2]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
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|
137 |
next |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
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|
138 |
case False |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
139 |
thus ?thesis |
14468 | 140 |
by (auto simp add: linorder_not_less dest: order_le_less_trans) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
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14371
diff
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|
141 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
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|
142 |
|
17433 | 143 |
subsection{*Properties of the set of embedded natural numbers*} |
14378
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generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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|
144 |
|
20740
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|
145 |
lemma of_nat_eq_star_of [simp]: "of_nat = star_of" |
5a103b43da5a
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|
146 |
proof |
5a103b43da5a
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|
147 |
fix n :: nat |
5a103b43da5a
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|
148 |
show "of_nat n = star_of n" by transfer simp |
5a103b43da5a
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|
149 |
qed |
5a103b43da5a
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|
150 |
|
5a103b43da5a
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|
151 |
lemma Nats_eq_Standard: "(Nats :: nat star set) = Standard" |
5a103b43da5a
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|
152 |
by (auto simp add: Nats_def Standard_def) |
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|
153 |
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lemma hypnat_of_nat_mem_Nats [simp]: "hypnat_of_nat n \<in> Nats" |
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by (simp add: Nats_eq_Standard) |
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|
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lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)" |
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by transfer simp |
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|
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lemma hypnat_of_nat_Suc [simp]: |
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"hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)" |
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by transfer simp |
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|
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lemma of_nat_eq_add [rule_format]: |
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"\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat" |
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apply (induct n) |
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apply (auto simp add: add_assoc) |
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apply (case_tac x) |
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apply (auto simp add: add_commute [of 1]) |
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done |
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|
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lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats" |
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by (simp add: Nats_eq_Standard) |
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|
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|
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subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*} |
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|
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definition |
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(* the set of infinite hypernatural numbers *) |
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HNatInfinite :: "hypnat set" where |
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"HNatInfinite = {n. n \<notin> Nats}" |
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|
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lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)" |
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by (simp add: HNatInfinite_def) |
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|
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lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)" |
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by (simp add: HNatInfinite_def) |
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|
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lemma star_of_neq_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<noteq> N" |
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by (auto simp add: HNatInfinite_def Nats_eq_Standard) |
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|
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lemma star_of_Suc_lessI: |
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"\<And>N. \<lbrakk>star_of n < N; star_of (Suc n) \<noteq> N\<rbrakk> \<Longrightarrow> star_of (Suc n) < N" |
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194 |
by transfer (rule Suc_lessI) |
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|
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196 |
lemma star_of_less_HNatInfinite: |
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197 |
assumes N: "N \<in> HNatInfinite" |
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198 |
shows "star_of n < N" |
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proof (induct n) |
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case 0 |
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from N have "star_of 0 \<noteq> N" by (rule star_of_neq_HNatInfinite) |
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thus "star_of 0 < N" by simp |
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next |
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case (Suc n) |
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from N have "star_of (Suc n) \<noteq> N" by (rule star_of_neq_HNatInfinite) |
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with Suc show "star_of (Suc n) < N" by (rule star_of_Suc_lessI) |
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207 |
qed |
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|
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lemma star_of_le_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<le> N" |
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by (rule star_of_less_HNatInfinite [THEN order_less_imp_le]) |
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|
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subsubsection {* Closure Rules *} |
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|
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lemma Nats_less_HNatInfinite: "\<lbrakk>x \<in> Nats; y \<in> HNatInfinite\<rbrakk> \<Longrightarrow> x < y" |
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by (auto simp add: Nats_def star_of_less_HNatInfinite) |
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|
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lemma Nats_le_HNatInfinite: "\<lbrakk>x \<in> Nats; y \<in> HNatInfinite\<rbrakk> \<Longrightarrow> x \<le> y" |
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by (rule Nats_less_HNatInfinite [THEN order_less_imp_le]) |
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|
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lemma zero_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 0 < x" |
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by (simp add: Nats_less_HNatInfinite) |
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|
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lemma one_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 < x" |
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by (simp add: Nats_less_HNatInfinite) |
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|
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lemma one_le_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 \<le> x" |
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by (simp add: Nats_le_HNatInfinite) |
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|
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lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite" |
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by (simp add: HNatInfinite_def) |
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|
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lemma Nats_downward_closed: |
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"\<lbrakk>x \<in> Nats; (y::hypnat) \<le> x\<rbrakk> \<Longrightarrow> y \<in> Nats" |
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apply (simp only: linorder_not_less [symmetric]) |
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apply (erule contrapos_np) |
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236 |
apply (drule HNatInfinite_not_Nats_iff [THEN iffD2]) |
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apply (erule (1) Nats_less_HNatInfinite) |
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238 |
done |
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239 |
|
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240 |
lemma HNatInfinite_upward_closed: |
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241 |
"\<lbrakk>x \<in> HNatInfinite; x \<le> y\<rbrakk> \<Longrightarrow> y \<in> HNatInfinite" |
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apply (simp only: HNatInfinite_not_Nats_iff) |
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apply (erule contrapos_nn) |
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apply (erule (1) Nats_downward_closed) |
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done |
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246 |
|
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lemma HNatInfinite_add: "x \<in> HNatInfinite \<Longrightarrow> x + y \<in> HNatInfinite" |
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apply (erule HNatInfinite_upward_closed) |
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apply (rule hypnat_le_add1) |
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250 |
done |
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|
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252 |
lemma HNatInfinite_add_one: "x \<in> HNatInfinite \<Longrightarrow> x + 1 \<in> HNatInfinite" |
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253 |
by (rule HNatInfinite_add) |
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254 |
|
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255 |
lemma HNatInfinite_diff: |
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256 |
"\<lbrakk>x \<in> HNatInfinite; y \<in> Nats\<rbrakk> \<Longrightarrow> x - y \<in> HNatInfinite" |
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apply (frule (1) Nats_le_HNatInfinite) |
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apply (simp only: HNatInfinite_not_Nats_iff) |
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apply (erule contrapos_nn) |
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apply (drule (1) Nats_add, simp) |
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261 |
done |
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262 |
|
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263 |
lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)" |
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apply (rule_tac x = "x - (1::hypnat) " in exI) |
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apply (simp add: Nats_le_HNatInfinite) |
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266 |
done |
17433 | 267 |
|
268 |
||
269 |
subsection{*Existence of an infinite hypernatural number*} |
|
270 |
||
19765 | 271 |
definition |
17433 | 272 |
(* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *) |
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whn :: hypnat where |
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hypnat_omega_def: "whn = star_n (%n::nat. n)" |
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|
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parents:
20730
diff
changeset
|
276 |
lemma hypnat_of_nat_neq_whn: "hypnat_of_nat n \<noteq> whn" |
21855
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
277 |
by (simp add: hypnat_omega_def star_of_def star_n_eq_iff) |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
278 |
|
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
279 |
lemma whn_neq_hypnat_of_nat: "whn \<noteq> hypnat_of_nat n" |
21855
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
280 |
by (simp add: hypnat_omega_def star_of_def star_n_eq_iff) |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
281 |
|
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
282 |
lemma whn_not_Nats [simp]: "whn \<notin> Nats" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
283 |
by (simp add: Nats_def image_def whn_neq_hypnat_of_nat) |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
284 |
|
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
285 |
lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
286 |
by (simp add: HNatInfinite_def) |
17433 | 287 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
288 |
lemma lemma_unbounded_set [simp]: "{n::nat. m < n} \<in> FreeUltrafilterNat" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
289 |
apply (insert finite_atMost [of m]) |
21855
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
290 |
apply (simp add: atMost_def) |
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
291 |
apply (drule FreeUltrafilterNat.finite) |
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
292 |
apply (drule FreeUltrafilterNat.not_memD) |
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
293 |
apply (simp add: Collect_neg_eq [symmetric] linorder_not_le) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
294 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
295 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
296 |
lemma Compl_Collect_le: "- {n::nat. N \<le> n} = {n. n < N}" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
297 |
by (simp add: Collect_neg_eq [symmetric] linorder_not_le) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
298 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
299 |
lemma hypnat_of_nat_eq: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
300 |
"hypnat_of_nat m = star_n (%n::nat. m)" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
301 |
by (simp add: star_of_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
302 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
303 |
lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
304 |
by (simp add: Nats_def image_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
305 |
|
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
306 |
lemma Nats_less_whn: "n \<in> Nats \<Longrightarrow> n < whn" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
307 |
by (simp add: Nats_less_HNatInfinite) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
308 |
|
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
309 |
lemma Nats_le_whn: "n \<in> Nats \<Longrightarrow> n \<le> whn" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
310 |
by (simp add: Nats_le_HNatInfinite) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
311 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
312 |
lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
313 |
by (simp add: Nats_less_whn) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
314 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
315 |
lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
316 |
by (simp add: Nats_le_whn) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
317 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
318 |
lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
319 |
by (simp add: Nats_less_whn) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
320 |
|
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
321 |
lemma hypnat_one_less_hypnat_omega [simp]: "1 < whn" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
322 |
by (simp add: Nats_less_whn) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
323 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
324 |
|
17433 | 325 |
subsubsection{*Alternative characterization of the set of infinite hypernaturals*} |
15070 | 326 |
|
327 |
text{* @{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}*} |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
328 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
329 |
(*??delete? similar reasoning in hypnat_omega_gt_SHNat above*) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
330 |
lemma HNatInfinite_FreeUltrafilterNat_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
331 |
"\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
332 |
==> {n. N < f n} \<in> FreeUltrafilterNat" |
15251 | 333 |
apply (induct_tac N) |
25162 | 334 |
apply (drule_tac x = 0 in spec, simp) |
21855
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
335 |
apply (drule_tac x = "Suc n" in spec) |
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
336 |
apply (elim ultra, auto) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
337 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
338 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
339 |
lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}" |
21855
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
340 |
apply (safe intro!: Nats_less_HNatInfinite) |
74909ecaf20a
remove ultra tactic and redundant FreeUltrafilterNat lemmas
huffman
parents:
21847
diff
changeset
|
341 |
apply (auto simp add: HNatInfinite_def) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
342 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
343 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
344 |
|
17433 | 345 |
subsubsection{*Alternative Characterization of @{term HNatInfinite} using |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
346 |
Free Ultrafilter*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
347 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
348 |
lemma HNatInfinite_FreeUltrafilterNat: |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
349 |
"star_n X \<in> HNatInfinite ==> \<forall>u. {n. u < X n}: FreeUltrafilterNat" |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
350 |
apply (auto simp add: HNatInfinite_iff SHNat_eq) |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
351 |
apply (drule_tac x="star_of u" in spec, simp) |
21865
55cc354fd2d9
moved several theorems; rearranged theory dependencies
huffman
parents:
21864
diff
changeset
|
352 |
apply (simp add: star_of_def star_less_def starP2_star_n) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
353 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
354 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
355 |
lemma FreeUltrafilterNat_HNatInfinite: |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
356 |
"\<forall>u. {n. u < X n}: FreeUltrafilterNat ==> star_n X \<in> HNatInfinite" |
21865
55cc354fd2d9
moved several theorems; rearranged theory dependencies
huffman
parents:
21864
diff
changeset
|
357 |
by (auto simp add: star_less_def starP2_star_n HNatInfinite_iff SHNat_eq hypnat_of_nat_eq) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
358 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
359 |
lemma HNatInfinite_FreeUltrafilterNat_iff: |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
360 |
"(star_n X \<in> HNatInfinite) = (\<forall>u. {n. u < X n}: FreeUltrafilterNat)" |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
361 |
by (rule iffI [OF HNatInfinite_FreeUltrafilterNat |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
362 |
FreeUltrafilterNat_HNatInfinite]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
363 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
364 |
subsection {* Embedding of the Hypernaturals into other types *} |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
365 |
|
19765 | 366 |
definition |
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
367 |
of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star" where |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
368 |
of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
369 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
370 |
lemma of_hypnat_0 [simp]: "of_hypnat 0 = 0" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
371 |
by transfer (rule of_nat_0) |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
372 |
|
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
373 |
lemma of_hypnat_1 [simp]: "of_hypnat 1 = 1" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
374 |
by transfer (rule of_nat_1) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
375 |
|
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
21865
diff
changeset
|
376 |
lemma of_hypnat_hSuc: "\<And>m. of_hypnat (hSuc m) = 1 + of_hypnat m" |
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
377 |
by transfer (rule of_nat_Suc) |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
378 |
|
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
379 |
lemma of_hypnat_add [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
380 |
"\<And>m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
381 |
by transfer (rule of_nat_add) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
382 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
383 |
lemma of_hypnat_mult [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
384 |
"\<And>m n. of_hypnat (m * n) = of_hypnat m * of_hypnat n" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
385 |
by transfer (rule of_nat_mult) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
386 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
387 |
lemma of_hypnat_less_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
388 |
"\<And>m n. (of_hypnat m < (of_hypnat n::'a::ordered_semidom star)) = (m < n)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
389 |
by transfer (rule of_nat_less_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
390 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
391 |
lemma of_hypnat_0_less_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
392 |
"\<And>n. (0 < (of_hypnat n::'a::ordered_semidom star)) = (0 < n)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
393 |
by transfer (rule of_nat_0_less_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
394 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
395 |
lemma of_hypnat_less_0_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
396 |
"\<And>m. \<not> (of_hypnat m::'a::ordered_semidom star) < 0" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
397 |
by transfer (rule of_nat_less_0_iff) |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
398 |
|
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
399 |
lemma of_hypnat_le_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
400 |
"\<And>m n. (of_hypnat m \<le> (of_hypnat n::'a::ordered_semidom star)) = (m \<le> n)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
401 |
by transfer (rule of_nat_le_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
402 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
403 |
lemma of_hypnat_0_le_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
404 |
"\<And>n. 0 \<le> (of_hypnat n::'a::ordered_semidom star)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
405 |
by transfer (rule of_nat_0_le_iff) |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
406 |
|
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
407 |
lemma of_hypnat_le_0_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
408 |
"\<And>m. ((of_hypnat m::'a::ordered_semidom star) \<le> 0) = (m = 0)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
409 |
by transfer (rule of_nat_le_0_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
410 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
411 |
lemma of_hypnat_eq_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
412 |
"\<And>m n. (of_hypnat m = (of_hypnat n::'a::ordered_semidom star)) = (m = n)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
413 |
by transfer (rule of_nat_eq_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
414 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
415 |
lemma of_hypnat_eq_0_iff [simp]: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
416 |
"\<And>m. ((of_hypnat m::'a::ordered_semidom star) = 0) = (m = 0)" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
417 |
by transfer (rule of_nat_eq_0_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
418 |
|
21864
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
419 |
lemma HNatInfinite_of_hypnat_gt_zero: |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
420 |
"N \<in> HNatInfinite \<Longrightarrow> (0::'a::ordered_semidom star) < of_hypnat N" |
2ecfd8985982
hypreal_of_hypnat abbreviates more general of_hypnat
huffman
parents:
21855
diff
changeset
|
421 |
by (rule ccontr, simp add: linorder_not_less) |
14420
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
422 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
423 |
end |