def low_r

bend-collections-laws-math@1.0.0.0 · proofs/math/random/pcg/step.bend:37

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/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

37def low_r(+k: Nat, +a: Nat, +b: Nat) -> {C.low(k, Nat.add(a, C.low(k, b))) == C.low(k, Nat.add(a, b)) : Nat}:
38  Equal.trans(Nat, C.low(k, Nat.add(a, C.low(k, b))), C.low(k, Nat.add(C.low(k, b), a)), C.low(k, Nat.add(a, b)),
39    Equal.cong(Nat, Nat, z => C.low(k, z), Nat.add(a, C.low(k, b)), Nat.add(C.low(k, b), a), NA.add_comm(a, C.low(k, b))),
40    Equal.trans(Nat, C.low(k, Nat.add(C.low(k, b), a)), C.low(k, Nat.add(b, a)), C.low(k, Nat.add(a, b)),
41      low_l(k, b, a),
42      Equal.cong(Nat, Nat, z => C.low(k, z), Nat.add(b, a), Nat.add(a, b), NA.add_comm(b, a))))