def diff_v

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/listops.bend:150

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
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend as LV

150def diff_v(+one: Nat, +h1: {one == 1n : Nat}, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +acc: Nat, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {Nat.is_eq(L.diff(xs, ys, acc), 0n) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(LV.ev(xs), LV.ev(ys))) : Bool}:
151  match n xs ys:
152    case 0n Nil{} Nil{}:
153      Equal.sym(Bool, Bool.and(Nat.is_eq(acc, 0n), True{}), Nat.is_eq(acc, 0n), and_true_r(Nat.is_eq(acc, 0n)))
154    case 0n Nil{} y <> yt:
155      Empty.absurd({Nat.is_eq(L.diff([], y <> yt, acc), 0n) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(LV.ev([]), LV.ev(y <> yt))) : Bool}, Lg.false_true(hy))
156    case 0n x <> xt _:
157      Empty.absurd({Nat.is_eq(L.diff(x <> xt, ys, acc), 0n) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(LV.ev(x <> xt), LV.ev(ys))) : Bool}, Lg.false_true(hx))
158    case 1n+k Nil{} _:
159      Empty.absurd({Nat.is_eq(L.diff([], ys, acc), 0n) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(LV.ev([]), LV.ev(ys))) : Bool}, Lg.false_true(hx))
160    case 1n+k x <> xt Nil{}:
161      Empty.absurd({Nat.is_eq(L.diff(x <> xt, [], acc), 0n) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(LV.ev(x <> xt), LV.ev([]))) : Bool}, Lg.false_true(hy))
162    case 1n+ +k +x <> +xt +y <> +yt:
163      +hx0 = Lg.and_left(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
164      +hy0 = Lg.and_left(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
165      +hxt = Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
166      +hyt = Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
167      %Equal.sym(Bool, Nat.is_eq(L.diff(xt, yt, Nat.add(acc, Nat.add(Nat.sub(x, y), Nat.sub(y, x)))), 0n), Bool.and(Nat.is_eq(Nat.add(acc, Nat.add(Nat.sub(x, y), Nat.sub(y, x))), 0n), Nat.is_eq(LV.ev(xt), LV.ev(yt))), diff_v(one, h1, k, xt, yt, Nat.add(acc, Nat.add(Nat.sub(x, y), Nat.sub(y, x))), hxt, hyt)) : {_ == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(Nat.add(x, C.shift(16n, LV.ev(xt))), Nat.add(y, C.shift(16n, LV.ev(yt))))) : Bool}
168      %Equal.sym(Bool, Nat.is_eq(Nat.add(acc, Nat.add(Nat.sub(x, y), Nat.sub(y, x))), 0n), Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(Nat.add(Nat.sub(x, y), Nat.sub(y, x)), 0n)), is_eq_add0(acc, Nat.add(Nat.sub(x, y), Nat.sub(y, x)))) : {Bool.and(_, Nat.is_eq(LV.ev(xt), LV.ev(yt))) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(Nat.add(x, C.shift(16n, LV.ev(xt))), Nat.add(y, C.shift(16n, LV.ev(yt))))) : Bool}
169      %Equal.sym(Bool, Nat.is_eq(Nat.add(Nat.sub(x, y), Nat.sub(y, x)), 0n), Nat.is_eq(x, y), absdiff(x, y)) : {Bool.and(Bool.and(Nat.is_eq(acc, 0n), _), Nat.is_eq(LV.ev(xt), LV.ev(yt))) == Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(Nat.add(x, C.shift(16n, LV.ev(xt))), Nat.add(y, C.shift(16n, LV.ev(yt))))) : Bool}
170      %Equal.sym(Bool, Nat.is_eq(Nat.add(x, C.shift(16n, LV.ev(xt))), Nat.add(y, C.shift(16n, LV.ev(yt)))), Bool.and(Nat.is_eq(x, y), Nat.is_eq(LV.ev(xt), LV.ev(yt))), eq_digit(one, h1, x, y, LV.ev(xt), LV.ev(yt), hx0, hy0)) : {Bool.and(Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(x, y)), Nat.is_eq(LV.ev(xt), LV.ev(yt))) == Bool.and(Nat.is_eq(acc, 0n), _) : Bool}
171      %Equal.sym(Bool, Bool.and(Bool.and(Nat.is_eq(acc, 0n), Nat.is_eq(x, y)), Nat.is_eq(LV.ev(xt), LV.ev(yt))), Bool.and(Nat.is_eq(acc, 0n), Bool.and(Nat.is_eq(x, y), Nat.is_eq(LV.ev(xt), LV.ev(yt)))), and_assoc(Nat.is_eq(acc, 0n), Nat.is_eq(x, y), Nat.is_eq(LV.ev(xt), LV.ev(yt)))) : {_ == Bool.and(Nat.is_eq(acc, 0n), Bool.and(Nat.is_eq(x, y), Nat.is_eq(LV.ev(xt), LV.ev(yt)))) : Bool}
172      {==}