1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Library/Function_Algebras.thy Fri Aug 20 17:48:30 2010 +0200
1.3 @@ -0,0 +1,207 @@
1.4 +(* Title: HOL/Library/Function_Algebras.thy
1.5 + Author: Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
1.6 +*)
1.7 +
1.8 +header {* Pointwise instantiation of functions to algebra type classes *}
1.9 +
1.10 +theory Function_Algebras
1.11 +imports Main
1.12 +begin
1.13 +
1.14 +text {* Pointwise operations *}
1.15 +
1.16 +instantiation "fun" :: (type, plus) plus
1.17 +begin
1.18 +
1.19 +definition
1.20 + "f + g = (\<lambda>x. f x + g x)"
1.21 +
1.22 +instance ..
1.23 +
1.24 +end
1.25 +
1.26 +instantiation "fun" :: (type, zero) zero
1.27 +begin
1.28 +
1.29 +definition
1.30 + "0 = (\<lambda>x. 0)"
1.31 +
1.32 +instance ..
1.33 +
1.34 +end
1.35 +
1.36 +instantiation "fun" :: (type, times) times
1.37 +begin
1.38 +
1.39 +definition
1.40 + "f * g = (\<lambda>x. f x * g x)"
1.41 +
1.42 +instance ..
1.43 +
1.44 +end
1.45 +
1.46 +instantiation "fun" :: (type, one) one
1.47 +begin
1.48 +
1.49 +definition
1.50 + "1 = (\<lambda>x. 1)"
1.51 +
1.52 +instance ..
1.53 +
1.54 +end
1.55 +
1.56 +
1.57 +text {* Additive structures *}
1.58 +
1.59 +instance "fun" :: (type, semigroup_add) semigroup_add proof
1.60 +qed (simp add: plus_fun_def add.assoc)
1.61 +
1.62 +instance "fun" :: (type, cancel_semigroup_add) cancel_semigroup_add proof
1.63 +qed (simp_all add: plus_fun_def expand_fun_eq)
1.64 +
1.65 +instance "fun" :: (type, ab_semigroup_add) ab_semigroup_add proof
1.66 +qed (simp add: plus_fun_def add.commute)
1.67 +
1.68 +instance "fun" :: (type, cancel_ab_semigroup_add) cancel_ab_semigroup_add proof
1.69 +qed simp
1.70 +
1.71 +instance "fun" :: (type, monoid_add) monoid_add proof
1.72 +qed (simp_all add: plus_fun_def zero_fun_def)
1.73 +
1.74 +instance "fun" :: (type, comm_monoid_add) comm_monoid_add proof
1.75 +qed simp
1.76 +
1.77 +instance "fun" :: (type, cancel_comm_monoid_add) cancel_comm_monoid_add ..
1.78 +
1.79 +instance "fun" :: (type, group_add) group_add proof
1.80 +qed (simp_all add: plus_fun_def zero_fun_def fun_Compl_def fun_diff_def diff_minus)
1.81 +
1.82 +instance "fun" :: (type, ab_group_add) ab_group_add proof
1.83 +qed (simp_all add: diff_minus)
1.84 +
1.85 +
1.86 +text {* Multiplicative structures *}
1.87 +
1.88 +instance "fun" :: (type, semigroup_mult) semigroup_mult proof
1.89 +qed (simp add: times_fun_def mult.assoc)
1.90 +
1.91 +instance "fun" :: (type, ab_semigroup_mult) ab_semigroup_mult proof
1.92 +qed (simp add: times_fun_def mult.commute)
1.93 +
1.94 +instance "fun" :: (type, ab_semigroup_idem_mult) ab_semigroup_idem_mult proof
1.95 +qed (simp add: times_fun_def)
1.96 +
1.97 +instance "fun" :: (type, monoid_mult) monoid_mult proof
1.98 +qed (simp_all add: times_fun_def one_fun_def)
1.99 +
1.100 +instance "fun" :: (type, comm_monoid_mult) comm_monoid_mult proof
1.101 +qed simp
1.102 +
1.103 +
1.104 +text {* Misc *}
1.105 +
1.106 +instance "fun" :: (type, "Rings.dvd") "Rings.dvd" ..
1.107 +
1.108 +instance "fun" :: (type, mult_zero) mult_zero proof
1.109 +qed (simp_all add: zero_fun_def times_fun_def)
1.110 +
1.111 +instance "fun" :: (type, mult_mono) mult_mono proof
1.112 +qed (auto simp add: zero_fun_def times_fun_def le_fun_def intro: mult_left_mono mult_right_mono)
1.113 +
1.114 +instance "fun" :: (type, mult_mono1) mult_mono1 proof
1.115 +qed (auto simp add: zero_fun_def times_fun_def le_fun_def intro: mult_mono1)
1.116 +
1.117 +instance "fun" :: (type, zero_neq_one) zero_neq_one proof
1.118 +qed (simp add: zero_fun_def one_fun_def expand_fun_eq)
1.119 +
1.120 +
1.121 +text {* Ring structures *}
1.122 +
1.123 +instance "fun" :: (type, semiring) semiring proof
1.124 +qed (simp_all add: plus_fun_def times_fun_def algebra_simps)
1.125 +
1.126 +instance "fun" :: (type, comm_semiring) comm_semiring proof
1.127 +qed (simp add: plus_fun_def times_fun_def algebra_simps)
1.128 +
1.129 +instance "fun" :: (type, semiring_0) semiring_0 ..
1.130 +
1.131 +instance "fun" :: (type, comm_semiring_0) comm_semiring_0 ..
1.132 +
1.133 +instance "fun" :: (type, semiring_0_cancel) semiring_0_cancel ..
1.134 +
1.135 +instance "fun" :: (type, comm_semiring_0_cancel) comm_semiring_0_cancel ..
1.136 +
1.137 +instance "fun" :: (type, semiring_1) semiring_1 ..
1.138 +
1.139 +lemma of_nat_fun:
1.140 + shows "of_nat n = (\<lambda>x::'a. of_nat n)"
1.141 +proof -
1.142 + have comp: "comp = (\<lambda>f g x. f (g x))"
1.143 + by (rule ext)+ simp
1.144 + have plus_fun: "plus = (\<lambda>f g x. f x + g x)"
1.145 + by (rule ext, rule ext) (fact plus_fun_def)
1.146 + have "of_nat n = (comp (plus (1::'b)) ^^ n) (\<lambda>x::'a. 0)"
1.147 + by (simp add: of_nat_def plus_fun zero_fun_def one_fun_def comp)
1.148 + also have "... = comp ((plus 1) ^^ n) (\<lambda>x::'a. 0)"
1.149 + by (simp only: comp_funpow)
1.150 + finally show ?thesis by (simp add: of_nat_def comp)
1.151 +qed
1.152 +
1.153 +instance "fun" :: (type, comm_semiring_1) comm_semiring_1 ..
1.154 +
1.155 +instance "fun" :: (type, semiring_1_cancel) semiring_1_cancel ..
1.156 +
1.157 +instance "fun" :: (type, comm_semiring_1_cancel) comm_semiring_1_cancel ..
1.158 +
1.159 +instance "fun" :: (type, semiring_char_0) semiring_char_0 proof
1.160 + from inj_of_nat have "inj (\<lambda>n (x::'a). of_nat n :: 'b)"
1.161 + by (rule inj_fun)
1.162 + then have "inj (\<lambda>n. of_nat n :: 'a \<Rightarrow> 'b)"
1.163 + by (simp add: of_nat_fun)
1.164 + then show "inj (of_nat :: nat \<Rightarrow> 'a \<Rightarrow> 'b)" .
1.165 +qed
1.166 +
1.167 +instance "fun" :: (type, ring) ring ..
1.168 +
1.169 +instance "fun" :: (type, comm_ring) comm_ring ..
1.170 +
1.171 +instance "fun" :: (type, ring_1) ring_1 ..
1.172 +
1.173 +instance "fun" :: (type, comm_ring_1) comm_ring_1 ..
1.174 +
1.175 +instance "fun" :: (type, ring_char_0) ring_char_0 ..
1.176 +
1.177 +
1.178 +text {* Ordereded structures *}
1.179 +
1.180 +instance "fun" :: (type, ordered_ab_semigroup_add) ordered_ab_semigroup_add proof
1.181 +qed (auto simp add: plus_fun_def le_fun_def intro: add_left_mono)
1.182 +
1.183 +instance "fun" :: (type, ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
1.184 +
1.185 +instance "fun" :: (type, ordered_ab_semigroup_add_imp_le) ordered_ab_semigroup_add_imp_le proof
1.186 +qed (simp add: plus_fun_def le_fun_def)
1.187 +
1.188 +instance "fun" :: (type, ordered_comm_monoid_add) ordered_comm_monoid_add ..
1.189 +
1.190 +instance "fun" :: (type, ordered_ab_group_add) ordered_ab_group_add ..
1.191 +
1.192 +instance "fun" :: (type, ordered_semiring) ordered_semiring ..
1.193 +
1.194 +instance "fun" :: (type, ordered_comm_semiring) ordered_comm_semiring ..
1.195 +
1.196 +instance "fun" :: (type, ordered_cancel_semiring) ordered_cancel_semiring ..
1.197 +
1.198 +instance "fun" :: (type, ordered_cancel_comm_semiring) ordered_cancel_comm_semiring ..
1.199 +
1.200 +instance "fun" :: (type, ordered_ring) ordered_ring ..
1.201 +
1.202 +instance "fun" :: (type, ordered_comm_ring) ordered_comm_ring ..
1.203 +
1.204 +
1.205 +lemmas func_plus = plus_fun_def
1.206 +lemmas func_zero = zero_fun_def
1.207 +lemmas func_times = times_fun_def
1.208 +lemmas func_one = one_fun_def
1.209 +
1.210 +end