def AP.and_id

bend-osdev-array@0.1.0.1 · PROOF.bend:928

import Base
import bend-osdev-array@0.1.0.1/array.bend as A

928# a word below 2^d is unchanged by the mask
929def AP.and_id(n: Nat, w: Word(n), +d: Nat, h: {Nat.is_lt(Word.to_nat(n, w), A.Arrays.slots(d)) == True{} : Bool}) -> {Word.and(n, w, AP.lm(n, d)) == w : Word(n)}:
930  match n:
931    case 0n:
932      match w:
933        case WNil{}:
934          {==}
935    case 1n+ +q:
936      match w d:
937        case WCon{False{}, +t} 0n:
938          +z = AP.val0(q, t, AP.double_zero(Word.to_nat(q, t), AP.lt_one(Nat.double(Word.to_nat(q, t)), h)))
939          AP.wcon_cong(q, False{}, Word.and(q, t, Word.zero(q)), t, Equal.trans(Word(q), Word.and(q, t, Word.zero(q)), Word.zero(q), t, AP.and_zero(q, t), Equal.sym(Word(q), t, Word.zero(q), z)))
940        case WCon{True{}, +t} 0n:
941          Empty.absurd({Word.and(1n+q, WCon{True{}, t}, AP.lm(1n+q, 0n)) == WCon{True{}, t} : Word(1n+q)}, AP.tf(Equal.trans(Bool, True{}, Nat.is_lt(Nat.double(Word.to_nat(q, t)), 0n), False{}, Equal.sym(Bool, Nat.is_lt(Nat.double(Word.to_nat(q, t)), 0n), True{}, h), AP.lt_zero(Nat.double(Word.to_nat(q, t))))))
942        case WCon{False{}, +t} 1n+ +e:
943          AP.wcon_cong(q, False{}, Word.and(q, t, AP.lm(q, e)), t, AP.and_id(q, t, e, AP.lt_bv_inv(False{}, Word.to_nat(q, t), A.Arrays.slots(e), h)))
944        case WCon{True{}, +t} 1n+ +e:
945          AP.wcon_cong(q, True{}, Word.and(q, t, AP.lm(q, e)), t, AP.and_id(q, t, e, AP.lt_bv_inv(True{}, Word.to_nat(q, t), A.Arrays.slots(e), h)))