def eq_rel

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/ed25519/pcodec.bend:163

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/eqf.bend as EQ
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/rel.bend as RL

163def eq_rel(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +x: List<&2, U32>, +y: List<&2, U32>, +a: Nat, +b: Nat, +rx: RL.R(pp, x, a), +ry: RL.R(pp, y, b), +ha: {Nat.mod(a, 1n+pp) == a : Nat}, +hb: {Nat.mod(b, 1n+pp) == b : Nat}) -> {F.eq(x, y) == Nat.is_eq(a, b) : Bool}:
164  +e1 = EQ.eq_val(one, h1, pp, hP, x, y, RL.r_o(pp, x, a, rx), RL.r_o(pp, y, b, ry))
165  +ea = Equal.trans(Nat, Nat.mod(FS.value(x), 1n+pp), Nat.mod(a, 1n+pp), a, RL.r_c(pp, x, a, rx), ha)
166  +eb = Equal.trans(Nat, Nat.mod(FS.value(y), 1n+pp), Nat.mod(b, 1n+pp), b, RL.r_c(pp, y, b, ry), hb)
167  Equal.trans(Bool, F.eq(x, y), Nat.is_eq(Nat.mod(FS.value(x), 1n+pp), Nat.mod(FS.value(y), 1n+pp)), Nat.is_eq(a, b), e1, Equal.trans(Bool, Nat.is_eq(Nat.mod(FS.value(x), 1n+pp), Nat.mod(FS.value(y), 1n+pp)), Nat.is_eq(a, Nat.mod(FS.value(y), 1n+pp)), Nat.is_eq(a, b), Equal.cong(Nat, Bool, z => Nat.is_eq(z, Nat.mod(FS.value(y), 1n+pp)), Nat.mod(FS.value(x), 1n+pp), a, ea), Equal.cong(Nat, Bool, z => Nat.is_eq(a, z), Nat.mod(FS.value(y), 1n+pp), b, eb)))