def snoc

bend-collections-laws-math@1.0.0.0 · proofs/math/number/prime.bend:87

import Base
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

87def snoc(k: Nat, +n: Nat, +d: Nat) -> {SN.nodiv(1n+k, n, d) == Bool.and(SN.nodiv(k, n, d), Bool.not(Nat.is_eq(Nat.mod(n, Nat.add(d, k)), 0n))) : Bool}:
88  match k:
89    case 0n:
90      %Equal.sym(Nat, Nat.add(d, 0n), d, A.add_zero(d)) : {SN.nodiv(1n, n, d) == Bool.and(True{}, Bool.not(Nat.is_eq(Nat.mod(n, _), 0n))) : Bool}
91      and_true(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)))
92    case 1n+ +m:
93      %Equal.sym(Nat, Nat.add(d, 1n+m), 1n+Nat.add(d, m), A.add_succ(d, m)) : {SN.nodiv(2n+m, n, d) == Bool.and(SN.nodiv(1n+m, n, d), Bool.not(Nat.is_eq(Nat.mod(n, _), 0n))) : Bool}
94      %Equal.sym(Bool, SN.nodiv(1n+m, n, 1n+d), Bool.and(SN.nodiv(m, n, 1n+d), Bool.not(Nat.is_eq(Nat.mod(n, Nat.add(1n+d, m)), 0n))), snoc(m, n, 1n+d)) : {Bool.and(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)), _) == Bool.and(SN.nodiv(1n+m, n, d), Bool.not(Nat.is_eq(Nat.mod(n, 1n+Nat.add(d, m)), 0n))) : Bool}
95      and_assoc(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)), SN.nodiv(m, n, 1n+d), Bool.not(Nat.is_eq(Nat.mod(n, 1n+Nat.add(d, m)), 0n)))