def eight_p_val

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/consts.bend:92

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
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/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/word.bend as WD
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/num.bend as M
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend as PL

92# value(8p limbs) == 8 p
93def eight_p_val(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}) -> {valo(one, F.eight_p()) == Nat.mul(8n, 1n+pp) : Nat}:
94  %Equal.sym(List<&2, U32>, F.eight_p(), Con{1896, rep(30n, 2040, Con{1016, Nil{}})}, eight_p_shape()) : {valo(one, _) == Nat.mul(8n, 1n+pp) : Nat}
95  +R = valo(one, rep(30n, 2040, Con{1016, Nil{}}))
96  +a = Nat.mul(one, v(1896))
97  +b = Nat.mul(one, 152n)
98  +c8 = Nat.mul(one, 8n)
99  +t = Nat.add(valo(one, Con{1016, Nil{}}), c8)
100  +k1 = Equal.trans(Nat, Nat.add(a, b), Nat.mul(one, Nat.add(v(1896), 152n)), Nat.mul(one, WD.sc(8n, 8n)), Equal.sym(Nat, Nat.mul(one, Nat.add(v(1896), 152n)), Nat.add(a, b), NA.mul_add_left(one, v(1896), 152n)), Equal.cong(Nat, Nat, z => Nat.mul(one, z), Nat.add(v(1896), 152n), WD.sc(8n, 8n), {==}))
101  +k2 = Equal.trans(Nat, Nat.add(a, b), Nat.mul(one, WD.sc(8n, 8n)), WD.sc(8n, c8), k1, Equal.sym(Nat, WD.sc(8n, c8), Nat.mul(one, WD.sc(8n, 8n)), PL.sc_mul_l(8n, one, 8n)))
102  +e1 = Equal.trans(Nat, Nat.add(Nat.add(a, C.shift(8n, R)), b), Nat.add(a, Nat.add(C.shift(8n, R), b)), Nat.add(Nat.add(a, b), C.shift(8n, R)), NA.add_assoc(a, C.shift(8n, R), b),
103    Equal.trans(Nat, Nat.add(a, Nat.add(C.shift(8n, R), b)), Nat.add(a, Nat.add(b, C.shift(8n, R))), Nat.add(Nat.add(a, b), C.shift(8n, R)), M.cong_r(a, Nat.add(C.shift(8n, R), b), Nat.add(b, C.shift(8n, R)), NA.add_comm(C.shift(8n, R), b)), Equal.sym(Nat, Nat.add(Nat.add(a, b), C.shift(8n, R)), Nat.add(a, Nat.add(b, C.shift(8n, R))), NA.add_assoc(a, b, C.shift(8n, R)))))
104  +e2 = M.cong_l(Nat.add(a, b), WD.sc(8n, c8), C.shift(8n, R), k2)
105  +e3 = Equal.trans(Nat, Nat.add(WD.sc(8n, c8), C.shift(8n, R)), WD.sc(8n, Nat.add(c8, R)), WD.sc(8n, Nat.add(R, c8)), M.sc_add(8n, c8, R), M.cong_sc(8n, Nat.add(c8, R), Nat.add(R, c8), NA.add_comm(c8, R)))
106  +e4 = M.cong_sc(8n, Nat.add(R, c8), LM.shn(30n, t), run8(one, 30n, Con{1016, Nil{}}))
107  +k3 = Equal.trans(Nat, t, Nat.add(Nat.mul(one, v(1016)), c8), Nat.mul(one, Nat.add(v(1016), 8n)), M.cong_l(valo(one, Con{1016, Nil{}}), Nat.mul(one, v(1016)), c8, N.add_zero(Nat.mul(one, v(1016)))), Equal.sym(Nat, Nat.mul(one, Nat.add(v(1016), 8n)), Nat.add(Nat.mul(one, v(1016)), c8), NA.mul_add_left(one, v(1016), 8n)))
108  +k4 = L.subst(Nat, o => {Nat.mul(o, Nat.add(v(1016), 8n)) == WD.sc(10n, o) : Nat}, 1n, one, Equal.sym(Nat, one, 1n, h1), {==})
109  +e5 = M.cong_sc(8n, LM.shn(30n, t), LM.shn(30n, WD.sc(10n, one)), Equal.cong(Nat, Nat, z => LM.shn(30n, z), t, WD.sc(10n, one), Equal.trans(Nat, t, Nat.mul(one, Nat.add(v(1016), 8n)), WD.sc(10n, one), k3, k4)))
110  +tot = Equal.trans(Nat, Nat.add(Nat.add(a, C.shift(8n, R)), b), Nat.add(Nat.add(a, b), C.shift(8n, R)), WD.sc(258n, one), e1,
111    Equal.trans(Nat, Nat.add(Nat.add(a, b), C.shift(8n, R)), Nat.add(WD.sc(8n, c8), C.shift(8n, R)), WD.sc(258n, one), e2,
112      Equal.trans(Nat, Nat.add(WD.sc(8n, c8), C.shift(8n, R)), WD.sc(8n, Nat.add(R, c8)), WD.sc(258n, one), e3,
113        Equal.trans(Nat, WD.sc(8n, Nat.add(R, c8)), C.shift(8n, LM.shn(30n, t)), WD.sc(258n, one), e4, e5))))
114  +V = valo(one, Con{1896, rep(30n, 2040, Con{1016, Nil{}})})
115  +tot2 = Equal.trans(Nat, Nat.add(V, 152n), Nat.add(V, b), Nat.add(Nat.mul(8n, 1n+pp), 152n), M.cong_r(V, 152n, b, Equal.sym(Nat, b, 152n, one_k(one, h1, 152n))), Equal.trans(Nat, Nat.add(V, b), WD.sc(258n, one), Nat.add(Nat.mul(8n, 1n+pp), 152n), tot, s258(one, h1, pp, hP)))
116  +tot3 = Equal.trans(Nat, Nat.add(152n, V), Nat.add(V, 152n), Nat.add(152n, Nat.mul(8n, 1n+pp)), NA.add_comm(152n, V), Equal.trans(Nat, Nat.add(V, 152n), Nat.add(Nat.mul(8n, 1n+pp), 152n), Nat.add(152n, Nat.mul(8n, 1n+pp)), tot2, NA.add_comm(Nat.mul(8n, 1n+pp), 152n)))
117  NR.add_cancel(152n, V, Nat.mul(8n, 1n+pp), tot3)