Reduce.lt_cancel_r: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/reduce.bend as Reduce
Lines 41-48 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/reduce.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/reduce.bend&start_line=41&end_line=48

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

# a + t < b + t -> a < b
def 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}:
  +h1 = N.lt_succ_le_succ(Nat.add(a, t), Nat.add(b, t), h)
  +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)
  +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)
  +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)
  +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)
  N.succ_le_lt(a, b, h5)

Uses (10):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_succ_le_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=succ_le_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=le_add_cancel