def lt_cancel_r

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/reduce.bend:41

import Base
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/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR

41# a + t < b + t -> a < b
42def lt_cancel_r(+a: Nat, +b: Nat, +t: Nat, +h: {Nat.is_lt(Nat.add(a, t), Nat.add(b, t)) == True{} : Bool}) -> {Nat.is_lt(a, b) == True{} : Bool}:
43  +h1 = N.lt_succ_le_succ(Nat.add(a, t), Nat.add(b, t), h)
44  +h2 = L.subst(Nat, z => {Nat.is_le(z, Nat.add(b, t)) == True{} : Bool}, 1n+Nat.add(a, t), Nat.add(1n+a, t), {==}, h1)
45  +h3 = L.subst(Nat, z => {Nat.is_le(z, Nat.add(b, t)) == True{} : Bool}, Nat.add(1n+a, t), Nat.add(t, 1n+a), NA.add_comm(1n+a, t), h2)
46  +h4 = L.subst(Nat, z => {Nat.is_le(Nat.add(t, 1n+a), z) == True{} : Bool}, Nat.add(b, t), Nat.add(t, b), NA.add_comm(b, t), h3)
47  +h5 = L.subst(Bool, z => {z == True{} : Bool}, Nat.is_le(Nat.add(t, 1n+a), Nat.add(t, b)), Nat.is_le(1n+a, b), NR.le_add_cancel(t, 1n+a, b), h4)
48  N.succ_le_lt(a, b, h5)