def contra

bend-collections-laws-math@1.0.0.0 · proofs/math/number/prime.bend:127

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/spec/math/number.bend as SN

127# e divides n with d <= e < n, d * d > n and no divisor in [2, d): impossible
128def contra(+n: Nat, +dq: Nat, +eq: Nat, +q: Nat, +hn: {n == Nat.mul(q, 2n+eq) : Nat}, +hen: {Nat.is_lt(2n+eq, n) == True{} : Bool}, +hdd: {Nat.is_lt(n, Nat.mul(2n+dq, 2n+dq)) == True{} : Bool}, +hs: {SN.nodiv(dq, n, 2n) == True{} : Bool}, +hde: {Nat.is_le(2n+dq, 2n+eq) == True{} : Bool}) -> Empty:
129  match q:
130    case 0n:
131      +hz = Equal.trans(Nat, n, Nat.mul(0n, 2n+eq), 0n, hn, mul0(2n+eq))
132      N.lt_zero_absurd(2n+eq, Equal.trans(Bool, Nat.is_lt(2n+eq, 0n), Nat.is_lt(2n+eq, n), True{}, Equal.cong(Nat, Bool, t => Nat.is_lt(2n+eq, t), 0n, n, Equal.sym(Nat, n, 0n, hz)), hen))
133    case 1n:
134      +he = Equal.trans(Nat, n, Nat.mul(1n, 2n+eq), 2n+eq, hn, mul1(2n+eq))
135      L.true_not_false(Nat.is_lt(2n+eq, 2n+eq), Equal.trans(Bool, Nat.is_lt(2n+eq, 2n+eq), Nat.is_lt(2n+eq, n), True{}, Equal.cong(Nat, Bool, t => Nat.is_lt(2n+eq, t), 2n+eq, n, Equal.sym(Nat, n, 2n+eq, he)), hen), N.lt_irrefl(2n+eq))
136    case 2n+ +qq:
137      c_two(n, dq, eq, qq, hn, hdd, hs, hde, Nat.is_lt(2n+qq, 2n+dq), {==})