def sb_hi

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/bitsrel.bend:79

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/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend as WW

79# above 16: the bits of the rest
80def sb_hi(i: Nat, +x: Nat, +r: Nat, +f16: {C.fits(16n, x) == True{} : Bool}) -> {sb(i, 16n, Nat.add(x, C.shift(16n, r))) == sb(i, 0n, r) : List<&2, Nat>}:
81  match i:
82    case 0n:
83      {==}
84    case 1n+ +j:
85      %Equal.sym(List<&2, Nat>, sb(j, 16n, Nat.add(x, C.shift(16n, r))), sb(j, 0n, r), sb_hi(j, x, r, f16)) : {C.bit(C.high(Nat.add(16n, j), Nat.add(x, C.shift(16n, r)))) <> _ == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
86      %Equal.sym(Nat, C.high(Nat.add(16n, j), Nat.add(x, C.shift(16n, r))), C.high(j, C.high(16n, Nat.add(x, C.shift(16n, r)))), WW.high_comp(j, 16n, Nat.add(x, C.shift(16n, r)))) : {C.bit(_) <> sb(j, 0n, r) == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
87      %Equal.sym(Nat, C.high(16n, Nat.add(x, C.shift(16n, r))), Nat.add(C.high(16n, x), r), WW.high_add_shift(16n, x, r)) : {C.bit(C.high(j, _)) <> sb(j, 0n, r) == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
88      %Equal.sym(Nat, C.high(16n, x), 0n, N.eq_from_is_eq(C.high(16n, x), 0n, f16)) : {C.bit(C.high(j, Nat.add(_, r))) <> sb(j, 0n, r) == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
89      {==}