42 lines
1.1 KiB
Plaintext
42 lines
1.1 KiB
Plaintext
section \<open> Multiplication Group Classes \<close>
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theory Groups_mult
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imports Main
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begin
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notation times (infixl "\<cdot>" 70)
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class group_mult = inverse + monoid_mult +
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assumes left_inverse: "inverse a \<cdot> a = 1"
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assumes multi_inverse_conv_div [simp]: "a \<cdot> (inverse b) = a / b"
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begin
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lemma div_conv_mult_inverse: "a / b = a \<cdot> (inverse b)"
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by simp
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sublocale mult: group times 1 inverse
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by standard (simp_all add: left_inverse)
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lemma diff_self [simp]: "a / a = 1"
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using mult.right_inverse by auto
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lemma mult_distrib_inverse [simp]: "(a * b) / b = a"
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by (metis local.mult_1_right local.multi_inverse_conv_div mult.right_inverse mult_assoc)
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end
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class ab_group_mult = comm_monoid_mult + group_mult
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begin
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lemma mult_distrib_inverse' [simp]: "(a * b) / a = b"
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using local.mult_distrib_inverse mult_commute by fastforce
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lemma inverse_distrib: "inverse (a * b) = (inverse a) * (inverse b)"
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by (simp add: local.mult.inverse_distrib_swap mult_commute)
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end
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abbreviation npower :: "'a::{power,inverse} \<Rightarrow> nat \<Rightarrow> 'a" ("(_\<^sup>-\<^sup>_)" [1000,999] 999)
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where "npower x n \<equiv> inverse (x ^ n)"
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end |