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(* Author: Florian Haftmann, TUM
*)
section \<open>Proof of concept for residue rings over int using type numerals\<close>
theory Residue_Ring
imports Main "HOL-Library.Type_Length"
begin
class len2 = len0 +
assumes len_ge_2 [iff]: "2 \ LENGTH('a)"
begin
subclass len
proof
show "0 < LENGTH('a)" using len_ge_2
by arith
qed
lemma len_not_eq_Suc_0 [simp]:
"LENGTH('a) \ Suc 0"
using len_ge_2 by arith
end
instance bit0 and bit1 :: (len) len2
by standard (simp_all add: Suc_le_eq)
quotient_type (overloaded) 'a residue_ring = int / "\k l. k mod int (LENGTH('a)) = l mod int (LENGTH('a::len2))"
by (auto intro!: equivpI reflpI sympI transpI)
context semiring_1
begin
lift_definition repr :: "'b::len2 residue_ring \ 'a"
is "\k. of_nat (nat (k mod int (LENGTH('b))))"
by simp
end
instantiation residue_ring :: (len2) comm_ring_1
begin
lift_definition zero_residue_ring :: "'a residue_ring"
is 0 .
lift_definition one_residue_ring :: "'a residue_ring"
is 1 .
lift_definition plus_residue_ring :: "'a residue_ring \ 'a residue_ring \ 'a residue_ring"
is plus
by (fact mod_add_cong)
lift_definition uminus_residue_ring :: "'a residue_ring \ 'a residue_ring"
is uminus
by (fact mod_minus_cong)
lift_definition minus_residue_ring :: "'a residue_ring \ 'a residue_ring \ 'a residue_ring"
is minus
by (fact mod_diff_cong)
lift_definition times_residue_ring :: "'a residue_ring \ 'a residue_ring \ 'a residue_ring"
is times
by (fact mod_mult_cong)
instance
by (standard; transfer) (simp_all add: algebra_simps mod_eq_0_iff_dvd)
end
context
includes lifting_syntax
begin
lemma [transfer_rule]:
"((\) ===> pcr_residue_ring) of_bool of_bool"
by (unfold of_bool_def [abs_def]; transfer_prover)
lemma [transfer_rule]:
"((=) ===> pcr_residue_ring) numeral numeral"
by (rule transfer_rule_numeral; transfer_prover)
lemma [transfer_rule]:
"((=) ===> pcr_residue_ring) int of_nat"
by (rule transfer_rule_of_nat; transfer_prover)
end
end
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