def take_drop_v

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/limbs.bend:137

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/secp256k1/limbs.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/typed/width.bend as WW

137# ev(xs) = ev(take n xs) + 2^(16 n) ev(drop n xs)
138def take_drop_v(n: Nat, xs: List<&2, Nat>) -> {ev(xs) == Nat.add(ev(L.take(n, xs)), pos(n, ev(L.drop(n, xs)))) : Nat}:
139  match n xs:
140    case 0n Nil{}:
141      {==}
142    case 0n x <> t:
143      {==}
144    case 1n+ +k Nil{}:
145      %Equal.sym(Nat, pos(k, 0n), 0n, pos_zero(k)) : {0n == Nat.add(0n, C.shift(16n, _)) : Nat}
146      %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {0n == Nat.add(0n, _) : Nat}
147      %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {0n == _ : Nat}
148      {==}
149    case 1n+ +k +x <> +t:
150      %Equal.sym(Nat, ev(t), Nat.add(ev(L.take(k, t)), pos(k, ev(L.drop(k, t)))), take_drop_v(k, t)) : {Nat.add(x, C.shift(16n, _)) == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
151      %Equal.sym(Nat, C.shift(16n, Nat.add(ev(L.take(k, t)), pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, ev(L.take(k, t))), C.shift(16n, pos(k, ev(L.drop(k, t))))), WW.shift_add(16n, ev(L.take(k, t)), pos(k, ev(L.drop(k, t))))) : {Nat.add(x, _) == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
152      %Equal.sym(Nat, Nat.add(x, Nat.add(C.shift(16n, ev(L.take(k, t))), C.shift(16n, pos(k, ev(L.drop(k, t)))))), Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))), NA.add_swap(x, C.shift(16n, ev(L.take(k, t))), C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {_ == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
153      %Equal.sym(Nat, Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x), NA.add_comm(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), _) == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
154      %Equal.sym(Nat, Nat.add(x, C.shift(16n, ev(L.take(k, t)))), Nat.add(C.shift(16n, ev(L.take(k, t))), x), NA.add_comm(x, C.shift(16n, ev(L.take(k, t))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x)) == Nat.add(_, C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
155      %Equal.sym(Nat, Nat.add(Nat.add(C.shift(16n, ev(L.take(k, t))), x), C.shift(16n, pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))), NA.add_assoc(C.shift(16n, ev(L.take(k, t))), x, C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x)) == _ : Nat}
156      %Equal.sym(Nat, Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x), NA.add_comm(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x)) == Nat.add(C.shift(16n, ev(L.take(k, t))), _) : Nat}
157      {==}