def lt_mul_l

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/poly1305/arith.bend:136

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/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend as WW
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/w64mul.bend as WM

136# a >= 1, b < c: a b < a c
137def lt_mul_l(+a: Nat, +b: Nat, +c: Nat, +ha: {Nat.is_le(1n, a) == True{} : Bool}, +h: {Nat.is_lt(b, c) == True{} : Bool}) -> {Nat.is_lt(Nat.mul(a, b), Nat.mul(a, c)) == True{} : Bool}:
138  +ab = Nat.mul(a, b)
139  +s1 = N.lt_le_trans(ab, 1n+ab, Nat.add(a, ab), N.lt_succ(ab), WW.le_add_r(1n, a, ab, ha))
140  +s2 = le_substl(Nat.mul(a, 1n+b), Nat.add(a, ab), Nat.mul(a, c), NA.mul_succ(a, b), WM.le_mul_r(a, 1n+b, c, N.lt_succ_le_succ(b, c, h)))
141  N.lt_le_trans(ab, Nat.add(a, ab), Nat.mul(a, c), s1, s2)