author | huffman |
Fri, 09 Sep 2005 19:34:22 +0200 | |
changeset 17318 | bc1c75855f3d |
parent 17299 | c6eecde058e4 |
child 17332 | 4910cf8c0cd2 |
permissions | -rw-r--r-- |
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(* Title : HyperNat.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Converted to Isar and polished by lcp |
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*) |
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header{*Construction of Hypernaturals using Ultrafilters*} |
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theory HyperNat |
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imports Star |
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begin |
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types hypnat = "nat star" |
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syntax hypnat_of_nat :: "nat => nat star" |
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translations "hypnat_of_nat" => "star_of :: nat => nat star" |
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consts whn :: hypnat |
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defs |
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(* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *) |
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hypnat_omega_def: "whn == star_n (%n::nat. n)" |
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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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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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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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lemma hypnat_diff_diff_left: "!!i j k. (i::hypnat) - j - k = i - (j+k)" |
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by transfer (rule diff_diff_left) |
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lemma hypnat_diff_commute: "!!i j k. (i::hypnat) - j - k = i-k-j" |
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by transfer (rule diff_commute) |
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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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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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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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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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lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)" |
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by transfer (rule diff_add_0) |
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subsection{*Hyperreal Multiplication*} |
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lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - n) * k = (m * k) - (n * k)" |
59 |
by transfer (rule diff_mult_distrib) |
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lemma hypnat_diff_mult_distrib2: "!!k m n. (k::hypnat) * (m - n) = (k * m) - (k * n)" |
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by transfer (rule diff_mult_distrib2) |
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subsection{*Properties of The @{text "\<le>"} Relation*} |
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lemma hypnat_le_zero_cancel [iff]: "!!n. (n \<le> (0::hypnat)) = (n = 0)" |
67 |
by transfer (rule le_0_eq) |
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lemma hypnat_mult_is_0 [simp]: "!!m n. (m*n = (0::hypnat)) = (m=0 | n=0)" |
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by transfer (rule mult_is_0) |
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lemma hypnat_diff_is_0_eq [simp]: "!!m n. ((m::hypnat) - n = 0) = (m \<le> n)" |
73 |
by transfer (rule diff_is_0_eq) |
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subsection{*Theorems for Ordering*} |
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lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)" |
80 |
by transfer (rule not_less0) |
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lemma hypnat_less_one [iff]: |
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"!!n. (n < (1::hypnat)) = (n=0)" |
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by transfer (rule less_one) |
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85 |
||
86 |
lemma hypnat_add_diff_inverse: "!!m n. ~ m<n ==> n+(m-n) = (m::hypnat)" |
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by transfer (rule add_diff_inverse) |
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17299 | 89 |
lemma hypnat_le_add_diff_inverse [simp]: "!!m n. n \<le> m ==> n+(m-n) = (m::hypnat)" |
90 |
by transfer (rule le_add_diff_inverse) |
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lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n\<le>m ==> (m-n)+n = (m::hypnat)" |
93 |
by transfer (rule le_add_diff_inverse2) |
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declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le] |
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96 |
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17299 | 97 |
lemma hypnat_le0 [iff]: "!!n. (0::hypnat) \<le> n" |
98 |
by transfer (rule le0) |
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17299 | 100 |
lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) \<le> n + x" |
101 |
by transfer (rule le_add2) |
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102 |
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103 |
lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)" |
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104 |
by (insert add_strict_left_mono [OF zero_less_one], auto) |
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105 |
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106 |
lemma hypnat_neq0_conv [iff]: "(n \<noteq> 0) = (0 < (n::hypnat))" |
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107 |
by (simp add: order_less_le) |
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108 |
|
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109 |
lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)" |
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110 |
by (auto simp add: linorder_not_less [symmetric]) |
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111 |
|
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112 |
lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::hypnat))" |
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113 |
apply safe |
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114 |
apply (rule_tac x = "n - (1::hypnat) " in exI) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
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115 |
apply (simp add: hypnat_gt_zero_iff) |
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paulson
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116 |
apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto) |
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117 |
done |
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118 |
|
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119 |
lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))" |
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120 |
by (simp add: linorder_not_le [symmetric] add_commute [of x]) |
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121 |
|
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122 |
lemma hypnat_diff_split: |
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123 |
"P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))" |
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124 |
-- {* elimination of @{text -} on @{text hypnat} *} |
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125 |
proof (cases "a<b" rule: case_split) |
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126 |
case True |
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paulson
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|
127 |
thus ?thesis |
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generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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changeset
|
128 |
by (auto simp add: hypnat_add_self_not_less order_less_imp_le |
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129 |
hypnat_diff_is_0_eq [THEN iffD2]) |
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130 |
next |
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131 |
case False |
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generic of_nat and of_int functions, and generalization of iszero
paulson
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|
132 |
thus ?thesis |
14468 | 133 |
by (auto simp add: linorder_not_less dest: order_le_less_trans) |
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|
134 |
qed |
69c4d5997669
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|
135 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
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|
136 |
|
15053 | 137 |
subsection{*The Embedding @{term hypnat_of_nat} Preserves @{text |
138 |
comm_ring_1} and Order Properties*} |
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139 |
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140 |
constdefs |
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141 |
|
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142 |
(* the set of infinite hypernatural numbers *) |
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143 |
HNatInfinite :: "hypnat set" |
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144 |
"HNatInfinite == {n. n \<notin> Nats}" |
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|
145 |
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146 |
|
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147 |
lemma hypnat_of_nat_def: "hypnat_of_nat m == of_nat m" |
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148 |
by (transfer star_of_nat_def) simp |
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starfun, starset, and other functions on NS types are now polymorphic;
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|
149 |
|
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150 |
lemma hypnat_of_nat_add: |
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151 |
"hypnat_of_nat ((z::nat) + w) = hypnat_of_nat z + hypnat_of_nat w" |
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152 |
by simp |
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153 |
|
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changeset
|
154 |
lemma hypnat_of_nat_mult: |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
155 |
"hypnat_of_nat (z * w) = hypnat_of_nat z * hypnat_of_nat w" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
156 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
157 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
158 |
lemma hypnat_of_nat_less_iff: |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
159 |
"(hypnat_of_nat z < hypnat_of_nat w) = (z < w)" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
160 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
161 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
162 |
lemma hypnat_of_nat_le_iff: |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
163 |
"(hypnat_of_nat z \<le> hypnat_of_nat w) = (z \<le> w)" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
164 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
165 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
166 |
lemma hypnat_of_nat_eq_iff: |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
167 |
"(hypnat_of_nat z = hypnat_of_nat w) = (z = w)" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
168 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
169 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
170 |
lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
171 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
172 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
173 |
lemma hypnat_of_nat_zero: "hypnat_of_nat 0 = 0" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
174 |
by simp |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
175 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
176 |
lemma hypnat_of_nat_zero_iff: "(hypnat_of_nat n = 0) = (n = 0)" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
177 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
178 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
179 |
lemma hypnat_of_nat_Suc [simp]: |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
180 |
"hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
181 |
by (simp add: hypnat_of_nat_def) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
182 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
183 |
lemma hypnat_of_nat_minus: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
184 |
"hypnat_of_nat ((j::nat) - k) = hypnat_of_nat j - hypnat_of_nat k" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
185 |
by simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
186 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
187 |
|
15070 | 188 |
subsection{*Existence of an infinite hypernatural number*} |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
189 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
190 |
text{*Existence of infinite number not corresponding to any natural number |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
191 |
follows because member @{term FreeUltrafilterNat} is not finite. |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
192 |
See @{text HyperDef.thy} for similar argument.*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
193 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
194 |
|
15070 | 195 |
subsection{*Properties of the set of embedded natural numbers*} |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
196 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
197 |
lemma of_nat_eq_add [rule_format]: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
198 |
"\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
199 |
apply (induct n) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
200 |
apply (auto simp add: add_assoc) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
201 |
apply (case_tac x) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
202 |
apply (auto simp add: add_commute [of 1]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
203 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
204 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
205 |
lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats" |
14468 | 206 |
by (auto simp add: of_nat_eq_add Nats_def split: hypnat_diff_split) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
207 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
208 |
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
|
209 |
apply (insert finite_atMost [of m]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
210 |
apply (simp add: atMost_def) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
211 |
apply (drule FreeUltrafilterNat_finite) |
14468 | 212 |
apply (drule FreeUltrafilterNat_Compl_mem, ultra) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
213 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
214 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
215 |
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
|
216 |
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
|
217 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
218 |
lemma hypnat_of_nat_eq: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
219 |
"hypnat_of_nat m = star_n (%n::nat. m)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
220 |
apply (induct m) |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
221 |
apply (simp_all add: star_n_zero_num star_n_one_num star_n_add) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
222 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
223 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
224 |
lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
225 |
by (force simp add: hypnat_of_nat_def Nats_def) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
226 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
227 |
lemma hypnat_omega_gt_SHNat: |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
228 |
"n \<in> Nats ==> n < whn" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
229 |
by (auto simp add: hypnat_of_nat_eq star_n_less hypnat_omega_def SHNat_eq) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
230 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
231 |
(* Infinite hypernatural not in embedded Nats *) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
232 |
lemma SHNAT_omega_not_mem [simp]: "whn \<notin> Nats" |
14468 | 233 |
by (blast dest: hypnat_omega_gt_SHNat) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
234 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
235 |
lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
236 |
apply (insert hypnat_omega_gt_SHNat [of "hypnat_of_nat n"]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
237 |
apply (simp add: hypnat_of_nat_def) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
238 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
239 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
240 |
lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
241 |
by (rule hypnat_of_nat_less_whn [THEN order_less_imp_le]) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
242 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
243 |
lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
244 |
by (simp add: hypnat_omega_gt_SHNat) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
245 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
246 |
lemma hypnat_one_less_hypnat_omega [simp]: "(1::hypnat) < whn" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
247 |
by (simp add: hypnat_omega_gt_SHNat) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
248 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
249 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
250 |
subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
251 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
252 |
lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
253 |
by (simp add: HNatInfinite_def) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
254 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
255 |
lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
256 |
by (simp add: HNatInfinite_def) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
257 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
258 |
lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
259 |
by (simp add: HNatInfinite_def) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
260 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
261 |
|
15070 | 262 |
subsection{*Alternative characterization of the set of infinite hypernaturals*} |
263 |
||
264 |
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
|
265 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
266 |
(*??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
|
267 |
lemma HNatInfinite_FreeUltrafilterNat_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
268 |
"\<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
|
269 |
==> {n. N < f n} \<in> FreeUltrafilterNat" |
15251 | 270 |
apply (induct_tac N) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
271 |
apply (drule_tac x = 0 in spec) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
272 |
apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem, drule FreeUltrafilterNat_Int, assumption, simp) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
273 |
apply (drule_tac x = "Suc n" in spec, ultra) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
274 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
275 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
276 |
lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
277 |
apply (auto simp add: HNatInfinite_def SHNat_eq hypnat_of_nat_eq) |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
278 |
apply (rule_tac x = x in star_cases) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
279 |
apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
280 |
simp add: star_n_less FreeUltrafilterNat_Compl_iff1 |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
281 |
star_n_eq_iff Collect_neg_eq [symmetric]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
282 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
283 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
284 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
285 |
subsection{*Alternative Characterization of @{term HNatInfinite} using |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
286 |
Free Ultrafilter*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
287 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
288 |
lemma HNatInfinite_FreeUltrafilterNat: |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
289 |
"x \<in> HNatInfinite |
17299 | 290 |
==> \<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
291 |
apply (cases x) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
292 |
apply (auto simp add: HNatInfinite_iff SHNat_eq hypnat_of_nat_eq) |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
293 |
apply (rule bexI [OF _ Rep_star_star_n], clarify) |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
294 |
apply (auto simp add: hypnat_of_nat_def star_n_less) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
295 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
296 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
297 |
lemma FreeUltrafilterNat_HNatInfinite: |
17299 | 298 |
"\<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
299 |
==> x \<in> HNatInfinite" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
300 |
apply (cases x) |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
301 |
apply (auto simp add: star_n_less 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
|
302 |
apply (drule spec, ultra, auto) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
303 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
304 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
305 |
lemma HNatInfinite_FreeUltrafilterNat_iff: |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
306 |
"(x \<in> HNatInfinite) = |
17299 | 307 |
(\<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
308 |
by (blast intro: HNatInfinite_FreeUltrafilterNat |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
309 |
FreeUltrafilterNat_HNatInfinite) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
310 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
311 |
lemma HNatInfinite_gt_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) < x" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
312 |
by (auto simp add: HNatInfinite_iff) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
313 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
314 |
lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
315 |
apply (auto simp add: HNatInfinite_iff) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
316 |
apply (drule_tac a = " (1::hypnat) " in equals0D) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
317 |
apply simp |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
318 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
319 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
320 |
lemma HNatInfinite_not_eq_zero: "x \<in> HNatInfinite ==> 0 < x" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
321 |
apply (drule HNatInfinite_gt_one) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
322 |
apply (auto simp add: order_less_trans [OF zero_less_one]) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
323 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
324 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
325 |
lemma HNatInfinite_ge_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) \<le> x" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
326 |
by (blast intro: order_less_imp_le HNatInfinite_gt_one) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
327 |
|
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 |
subsection{*Closure Rules*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
330 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
331 |
lemma HNatInfinite_add: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
332 |
"[| x \<in> HNatInfinite; y \<in> HNatInfinite |] ==> x + y \<in> HNatInfinite" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
333 |
apply (auto simp add: HNatInfinite_iff) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
334 |
apply (drule bspec, assumption) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
335 |
apply (drule bspec [OF _ Nats_0]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
336 |
apply (drule add_strict_mono, assumption, simp) |
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 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
339 |
lemma HNatInfinite_SHNat_add: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
340 |
"[| x \<in> HNatInfinite; y \<in> Nats |] ==> x + y \<in> HNatInfinite" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
341 |
apply (auto simp add: HNatInfinite_not_Nats_iff) |
14468 | 342 |
apply (drule_tac a = "x + y" in Nats_diff, auto) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
343 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
344 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
345 |
lemma HNatInfinite_Nats_imp_less: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> y < x" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
346 |
by (simp add: HNatInfinite_iff) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
347 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
348 |
lemma HNatInfinite_SHNat_diff: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
349 |
assumes x: "x \<in> HNatInfinite" and y: "y \<in> Nats" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
350 |
shows "x - y \<in> HNatInfinite" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
351 |
proof - |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
352 |
have "y < x" by (simp add: HNatInfinite_Nats_imp_less prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
353 |
hence "x - y + y = x" by (simp add: order_less_imp_le) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
354 |
with x show ?thesis |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
355 |
by (force simp add: HNatInfinite_not_Nats_iff |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
356 |
dest: Nats_add [of "x-y", OF _ y]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
357 |
qed |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
358 |
|
14415 | 359 |
lemma HNatInfinite_add_one: |
360 |
"x \<in> HNatInfinite ==> x + (1::hypnat) \<in> HNatInfinite" |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
361 |
by (auto intro: HNatInfinite_SHNat_add) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
362 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
363 |
lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
364 |
apply (rule_tac x = "x - (1::hypnat) " in exI) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
365 |
apply auto |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
366 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
367 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
368 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
369 |
subsection{*Embedding of the Hypernaturals into the Hyperreals*} |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
370 |
text{*Obtained using the nonstandard extension of the naturals*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
371 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
372 |
constdefs |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
373 |
hypreal_of_hypnat :: "hypnat => hypreal" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
374 |
"hypreal_of_hypnat == *f* real" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
375 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
376 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
377 |
lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
378 |
by (simp add: hypreal_of_nat_def) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
379 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
380 |
lemma hypreal_of_hypnat: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
381 |
"hypreal_of_hypnat (star_n X) = star_n (%n. real (X n))" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
382 |
by (simp add: hypreal_of_hypnat_def starfun) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
383 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
384 |
lemma hypreal_of_hypnat_inject [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
385 |
"!!m n. (hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
386 |
by (unfold hypreal_of_hypnat_def, transfer, simp) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
387 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
388 |
lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
389 |
by (simp add: star_n_zero_num hypreal_of_hypnat) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
390 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
391 |
lemma hypreal_of_hypnat_one: "hypreal_of_hypnat (1::hypnat) = 1" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
392 |
by (simp add: star_n_one_num hypreal_of_hypnat) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
393 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
394 |
lemma hypreal_of_hypnat_add [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
395 |
"!!m n. hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
396 |
by (unfold hypreal_of_hypnat_def, transfer, rule real_of_nat_add) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
397 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
398 |
lemma hypreal_of_hypnat_mult [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
399 |
"!!m n. hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
400 |
by (unfold hypreal_of_hypnat_def, transfer, rule real_of_nat_mult) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
401 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
402 |
lemma hypreal_of_hypnat_less_iff [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
403 |
"!!m n. (hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
404 |
by (unfold hypreal_of_hypnat_def, transfer, simp) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
405 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
406 |
lemma hypreal_of_hypnat_eq_zero_iff: "(hypreal_of_hypnat N = 0) = (N = 0)" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
407 |
by (simp add: hypreal_of_hypnat_zero [symmetric]) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
408 |
declare hypreal_of_hypnat_eq_zero_iff [simp] |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
409 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
410 |
lemma hypreal_of_hypnat_ge_zero [simp]: "!!n. 0 \<le> hypreal_of_hypnat n" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
411 |
by (unfold hypreal_of_hypnat_def, transfer, simp) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
412 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
413 |
lemma HNatInfinite_inverse_Infinitesimal [simp]: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
414 |
"n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
415 |
apply (cases n) |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
416 |
apply (auto simp add: hypreal_of_hypnat star_n_inverse |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
417 |
HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2) |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
418 |
apply (rule bexI [OF _ Rep_star_star_n], auto) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
419 |
apply (drule_tac x = "m + 1" in spec, ultra) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
420 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
421 |
|
14420
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
422 |
lemma HNatInfinite_hypreal_of_hypnat_gt_zero: |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
423 |
"N \<in> HNatInfinite ==> 0 < hypreal_of_hypnat N" |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
424 |
apply (rule ccontr) |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
425 |
apply (simp add: hypreal_of_hypnat_zero [symmetric] linorder_not_less) |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
426 |
done |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
427 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
428 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
429 |
ML |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
430 |
{* |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
431 |
val hypnat_of_nat_def = thm"hypnat_of_nat_def"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
432 |
val HNatInfinite_def = thm"HNatInfinite_def"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
433 |
val hypreal_of_hypnat_def = thm"hypreal_of_hypnat_def"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
434 |
val hypnat_omega_def = thm"hypnat_omega_def"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
435 |
|
17299 | 436 |
val starrel_iff = thm "starrel_iff"; |
437 |
val lemma_starrel_refl = thm "lemma_starrel_refl"; |
|
438 |
val eq_Abs_star = thm "eq_Abs_star"; |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
439 |
val hypnat_minus_zero = thm "hypnat_minus_zero"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
440 |
val hypnat_diff_0_eq_0 = thm "hypnat_diff_0_eq_0"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
441 |
val hypnat_add_is_0 = thm "hypnat_add_is_0"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
442 |
val hypnat_diff_diff_left = thm "hypnat_diff_diff_left"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
443 |
val hypnat_diff_commute = thm "hypnat_diff_commute"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
444 |
val hypnat_diff_add_inverse = thm "hypnat_diff_add_inverse"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
445 |
val hypnat_diff_add_inverse2 = thm "hypnat_diff_add_inverse2"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
446 |
val hypnat_diff_cancel = thm "hypnat_diff_cancel"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
447 |
val hypnat_diff_cancel2 = thm "hypnat_diff_cancel2"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
448 |
val hypnat_diff_add_0 = thm "hypnat_diff_add_0"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
449 |
val hypnat_diff_mult_distrib = thm "hypnat_diff_mult_distrib"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
450 |
val hypnat_diff_mult_distrib2 = thm "hypnat_diff_mult_distrib2"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
451 |
val hypnat_mult_is_0 = thm "hypnat_mult_is_0"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
452 |
val hypnat_not_less0 = thm "hypnat_not_less0"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
453 |
val hypnat_less_one = thm "hypnat_less_one"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
454 |
val hypnat_add_diff_inverse = thm "hypnat_add_diff_inverse"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
455 |
val hypnat_le_add_diff_inverse = thm "hypnat_le_add_diff_inverse"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
456 |
val hypnat_le_add_diff_inverse2 = thm "hypnat_le_add_diff_inverse2"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
457 |
val hypnat_le0 = thm "hypnat_le0"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
458 |
val hypnat_add_self_le = thm "hypnat_add_self_le"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
459 |
val hypnat_add_one_self_less = thm "hypnat_add_one_self_less"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
460 |
val hypnat_neq0_conv = thm "hypnat_neq0_conv"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
461 |
val hypnat_gt_zero_iff = thm "hypnat_gt_zero_iff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
462 |
val hypnat_gt_zero_iff2 = thm "hypnat_gt_zero_iff2"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
463 |
val hypnat_of_nat_add = thm "hypnat_of_nat_add"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
464 |
val hypnat_of_nat_minus = thm "hypnat_of_nat_minus"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
465 |
val hypnat_of_nat_mult = thm "hypnat_of_nat_mult"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
466 |
val hypnat_of_nat_less_iff = thm "hypnat_of_nat_less_iff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
467 |
val hypnat_of_nat_le_iff = thm "hypnat_of_nat_le_iff"; |
14415 | 468 |
val hypnat_of_nat_eq = thm"hypnat_of_nat_eq" |
469 |
val SHNat_eq = thm"SHNat_eq" |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
470 |
val hypnat_of_nat_one = thm "hypnat_of_nat_one"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
471 |
val hypnat_of_nat_zero = thm "hypnat_of_nat_zero"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
472 |
val hypnat_of_nat_zero_iff = thm "hypnat_of_nat_zero_iff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
473 |
val hypnat_of_nat_Suc = thm "hypnat_of_nat_Suc"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
474 |
val SHNAT_omega_not_mem = thm "SHNAT_omega_not_mem"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
475 |
val cofinite_mem_FreeUltrafilterNat = thm "cofinite_mem_FreeUltrafilterNat"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
476 |
val hypnat_omega_gt_SHNat = thm "hypnat_omega_gt_SHNat"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
477 |
val hypnat_of_nat_less_whn = thm "hypnat_of_nat_less_whn"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
478 |
val hypnat_of_nat_le_whn = thm "hypnat_of_nat_le_whn"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
479 |
val hypnat_zero_less_hypnat_omega = thm "hypnat_zero_less_hypnat_omega"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
480 |
val hypnat_one_less_hypnat_omega = thm "hypnat_one_less_hypnat_omega"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
481 |
val HNatInfinite_whn = thm "HNatInfinite_whn"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
482 |
val HNatInfinite_iff = thm "HNatInfinite_iff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
483 |
val HNatInfinite_FreeUltrafilterNat = thm "HNatInfinite_FreeUltrafilterNat"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
484 |
val FreeUltrafilterNat_HNatInfinite = thm "FreeUltrafilterNat_HNatInfinite"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
485 |
val HNatInfinite_FreeUltrafilterNat_iff = thm "HNatInfinite_FreeUltrafilterNat_iff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
486 |
val HNatInfinite_gt_one = thm "HNatInfinite_gt_one"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
487 |
val zero_not_mem_HNatInfinite = thm "zero_not_mem_HNatInfinite"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
488 |
val HNatInfinite_not_eq_zero = thm "HNatInfinite_not_eq_zero"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
489 |
val HNatInfinite_ge_one = thm "HNatInfinite_ge_one"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
490 |
val HNatInfinite_add = thm "HNatInfinite_add"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
491 |
val HNatInfinite_SHNat_add = thm "HNatInfinite_SHNat_add"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
492 |
val HNatInfinite_SHNat_diff = thm "HNatInfinite_SHNat_diff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
493 |
val HNatInfinite_add_one = thm "HNatInfinite_add_one"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
494 |
val HNatInfinite_is_Suc = thm "HNatInfinite_is_Suc"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
495 |
val HNat_hypreal_of_nat = thm "HNat_hypreal_of_nat"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
496 |
val hypreal_of_hypnat = thm "hypreal_of_hypnat"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
497 |
val hypreal_of_hypnat_zero = thm "hypreal_of_hypnat_zero"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
498 |
val hypreal_of_hypnat_one = thm "hypreal_of_hypnat_one"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
499 |
val hypreal_of_hypnat_add = thm "hypreal_of_hypnat_add"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
500 |
val hypreal_of_hypnat_mult = thm "hypreal_of_hypnat_mult"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
501 |
val hypreal_of_hypnat_less_iff = thm "hypreal_of_hypnat_less_iff"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
502 |
val hypreal_of_hypnat_ge_zero = thm "hypreal_of_hypnat_ge_zero"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
503 |
val HNatInfinite_inverse_Infinitesimal = thm "HNatInfinite_inverse_Infinitesimal"; |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
504 |
*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
505 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
506 |
end |