def c_small

bend-collections-laws-math@1.0.0.0 · proofs/math/number/prime.bend:105

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/spec/math/number.bend as SN
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend as R

105# n == q e with q >= 2 and q < d would divide n, which nodiv(dq, n, 2) rules out
106def c_small(+n: Nat, +dq: Nat, +eq: Nat, +qq: Nat, +hn: {n == Nat.mul(2n+qq, 2n+eq) : Nat}, +hs: {SN.nodiv(dq, n, 2n) == True{} : Bool}, +hq: {Nat.is_lt(2n+qq, 2n+dq) == True{} : Bool}) -> Empty:
107  +hf = nodiv_at(dq, n, 2n, 2n+qq, hs, N.zero_le(qq), hq)
108  +hm = Equal.trans(Nat, Nat.mod(n, 2n+qq), Nat.mod(Nat.add(Nat.mul(2n+eq, 2n+qq), 0n), 2n+qq), 0n, Equal.cong(Nat, Nat, t => Nat.mod(t, 2n+qq), n, Nat.add(Nat.mul(2n+eq, 2n+qq), 0n), Equal.trans(Nat, n, Nat.mul(2n+qq, 2n+eq), Nat.add(Nat.mul(2n+eq, 2n+qq), 0n), hn, Equal.trans(Nat, Nat.mul(2n+qq, 2n+eq), Nat.mul(2n+eq, 2n+qq), Nat.add(Nat.mul(2n+eq, 2n+qq), 0n), A.mul_comm(2n+qq, 2n+eq), Equal.sym(Nat, Nat.add(Nat.mul(2n+eq, 2n+qq), 0n), Nat.mul(2n+eq, 2n+qq), A.add_zero(Nat.mul(2n+eq, 2n+qq)))))), R.mod_of(2n+eq, 1n+qq, 0n, {==}))
109  +ht = Equal.cong(Nat, Bool, t => Nat.is_eq(t, 0n), Nat.mod(n, 2n+qq), 0n, hm)
110  L.true_not_false(Nat.is_eq(Nat.mod(n, 2n+qq), 0n), ht, hf)