def sel_one

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/reduce.bend:156

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/spec/crypto/secp256k1/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/limbs.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as Lg

156# select(1, xs, ys) == xs and select(0, xs, ys) == ys for lists of n limbs
157def sel_one(+one: Nat, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {L.select(1n, xs, ys) == xs : List<&2, Nat>}:
158  match n xs ys:
159    case 0n Nil{} _:
160      {==}
161    case 0n x <> xt _:
162      Empty.absurd({L.select(1n, x <> xt, ys) == x <> xt : List<&2, Nat>}, Lg.false_true(hx))
163    case 1n+k Nil{} _:
164      Empty.absurd({L.select(1n, [], ys) == [] : List<&2, Nat>}, Lg.false_true(hx))
165    case 1n+k x <> xt Nil{}:
166      Empty.absurd({L.select(1n, x <> xt, []) == x <> xt : List<&2, Nat>}, Lg.false_true(hy))
167    case 1n+ +k +x <> +xt +y <> +yt:
168      +e1 = sel_one(one, k, xt, yt, Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx), Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy))
169      %Equal.sym(List<&2, Nat>, L.select(1n, xt, yt), xt, e1) : {L.pick1(1n, x, y) <> _ == x <> xt : List<&2, Nat>}
170      %Equal.sym(Nat, L.pick1(1n, x, y), x, pick_one(x, y)) : {_ <> xt == x <> xt : List<&2, Nat>}
171      {==}