def w_zero

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/eqf.bend:26

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend as G

26def w_zero(+pp: Nat, +A: Nat, +B: Nat, +hB: {Nat.is_lt(B, 1n+pp) == True{} : Bool}, +e: {A == B : Nat}) -> {Nat.mod(pw(pp, A, B), 1n+pp) == 0n : Nat}:
27  +P = {1n+pp : Nat}
28  +w = pw(pp, A, B)
29  +e2 = L.subst(Nat, z => {Nat.add(w, z) == Nat.add(A, P) : Nat}, B, A, Equal.sym(Nat, A, B, e), w_eq(pp, A, B, hB))
30  +e3 = NR.add_cancel(A, w, P, Equal.trans(Nat, Nat.add(A, w), Nat.add(w, A), Nat.add(A, P), NA.add_comm(A, w), e2))
31  +e4 = Equal.trans(Nat, P, Nat.mul(1n, P), Nat.add(Nat.mul(1n, P), 0n), Equal.sym(Nat, Nat.mul(1n, P), P, NA.add_zero(P)), Equal.sym(Nat, Nat.add(Nat.mul(1n, P), 0n), Nat.mul(1n, P), N.add_zero(Nat.mul(1n, P))))
32  Equal.trans(Nat, Nat.mod(w, P), Nat.mod(Nat.add(Nat.mul(1n, P), 0n), P), 0n, G.c_eq(pp, w, Nat.add(Nat.mul(1n, P), 0n), Equal.trans(Nat, w, P, Nat.add(Nat.mul(1n, P), 0n), e3, e4)), NR.mod_of(1n, pp, 0n, {==}))