def sm_v

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqr.bend:270

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA

270def sm_v(+one: Nat, +h1: {one == 1n : Nat}, +nh: WU.U64, +h62: {C.fits(62n, SW.value(nh)) == True{} : Bool}, +h60: {C.fits(60n, SW.value(nh)) == False{} : Bool}, +q: U32, +u: U32, +hc: {Nat.add(Nat.mul(v(q), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), v(u)) == C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))) : Nat}, +hU: {Nat.is_le(v(u), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))) == True{} : Bool}, +hQ: {Nat.is_lt(v(q), C.shift(32n, one)) == True{} : Bool}, +hBA: {Nat.is_lt(C.shift(32n, v(u)), Nat.mul(v(q), v(q))) == True{} : Bool}) -> {X.le(X.sub(X.mul32(q, q), WU.U64{0, u}), X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0})) == Nat.is_le(Nat.sub(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), Nat.sub(Nat.add(2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n), 2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n)), 1n)) : Bool}:
271  Equal.trans(Bool, X.le(X.sub(X.mul32(q, q), WU.U64{0, u}), X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0})), Nat.is_le(SW.value(X.sub(X.mul32(q, q), WU.U64{0, u})), SW.value(X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0}))), Nat.is_le(Nat.sub(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), Nat.sub(Nat.add(2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n), 2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n)), 1n)), WA.le_value(X.sub(X.mul32(q, q), WU.U64{0, u}), X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0})), Equal.trans(Bool, Nat.is_le(SW.value(X.sub(X.mul32(q, q), WU.U64{0, u})), SW.value(X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0}))), Nat.is_le(Nat.sub(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), SW.value(X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0}))), Nat.is_le(Nat.sub(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), Nat.sub(Nat.add(2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n), 2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n)), 1n)), Equal.cong(Nat, Bool, z => Nat.is_le(z, SW.value(X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0}))), SW.value(X.sub(X.mul32(q, q), WU.U64{0, u})), Nat.sub(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), dw_value(one, h1, nh, h62, h60, q, u, hc, hU, hQ, hBA)), Equal.cong(Nat, Bool, z => Nat.is_le(Nat.sub(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), z), SW.value(X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0})), Nat.sub(Nat.add(2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n), 2n+Nat.sub(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), 2n)), 1n), cw_v(one, h1, nh, h62, h60, q, u, hc, hU, hQ))))