def z_case

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/freeze.bend:88

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/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/u32.bend as U
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/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fold.bend as FD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/reduce.bend as RD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend as K

88# case c == 0 (no carry out): x < p, the result is x
89def z_case(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +x: List<&2, U32>, +cp: List<&2, U32>, +hx: {LM.okb(32n, x, 255n) == True{} : Bool}, +hk: {LM.okb(32n, cp, 128n) == True{} : Bool}, +hv: {K.valo(one, cp) == Nat.add(1n+pp, 38n) : Nat}, +hc: {Nat.is_lt(v(LM.cc(F.carry(F.addl(x, cp), 0))), one) == True{} : Bool}) -> {Nat.is_lt(FS.value(x), 1n+pp) == True{} : Bool}:
90  +P = {1n+pp : Nat}
91  +r = F.carry(F.addl(x, cp), 0)
92  +c = v(LM.cc(r))
93  +vs = FS.value(LM.cl(r))
94  +c0 = U.lt_one_zero(c, L.subst(Nat, o => {Nat.is_lt(c, o) == True{} : Bool}, one, 1n, h1, hc))
95  +ev = cs_val(one, h1, pp, x, cp, hx, hk, hv)
96  +ev0 = L.subst(Nat, z => {Nat.add(vs, LM.shn(32n, z)) == Nat.add(FS.value(x), Nat.add(P, 38n)) : Nat}, c, 0n, c0, ev)
97  +ev1 = Equal.trans(Nat, vs, Nat.add(vs, 0n), Nat.add(FS.value(x), Nat.add(P, 38n)), Equal.sym(Nat, Nat.add(vs, 0n), vs, N.add_zero(vs)), ev0)
98  +lt = RD.val_lt(one, h1, 32n, LM.cl(r), FD.cr_okb(one, h1, F.addl(x, cp), ad_okb(one, h1, x, cp, hx, hk)))
99  +lt2 = L.subst(Nat, z => {Nat.is_lt(vs, z) == True{} : Bool}, LM.shn(32n, one), Nat.add(Nat.add(P, P), 38n), t_val(one, h1, pp, hP), lt)
100  +lt3 = L.subst(Nat, z => {Nat.is_lt(z, Nat.add(Nat.add(P, P), 38n)) == True{} : Bool}, vs, Nat.add(FS.value(x), Nat.add(P, 38n)), ev1, lt2)
101  +lt4 = L.subst(Nat, z => {Nat.is_lt(Nat.add(FS.value(x), Nat.add(P, 38n)), z) == True{} : Bool}, Nat.add(Nat.add(P, P), 38n), Nat.add(P, Nat.add(P, 38n)), NA.add_assoc(P, P, 38n), lt3)
102  RD.lt_cancel_r(FS.value(x), P, Nat.add(P, 38n), lt4)