Reduce.mul_lt_cancel: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/reduce.bend as Reduce
Lines 74-81 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/reduce.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/reduce.bend&start_line=74&end_line=81

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 Lg
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend as AR

# T a < T b gives a < b
def mul_lt_cancel(+t: Nat, +a: Nat, +b: Nat, +h: {Nat.is_lt(Nat.mul(a, t), Nat.mul(b, t)) == True{} : Bool}, +c: Bool, +hc: {Nat.is_lt(a, b) == c : Bool}) -> {Nat.is_lt(a, b) == True{} : Bool}:
  match c:
    case True{}:
      hc
    case False{}:
      +le = AR.mul_le(b, a, t, N.not_lt_le(a, b, hc))
      Empty.absurd({Nat.is_lt(a, b) == True{} : Bool}, Lg.true_not_false(Nat.is_lt(Nat.mul(a, t), Nat.mul(b, t)), h, N.le_not_lt(Nat.mul(a, t), Nat.mul(b, t), le)))

Uses (9):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&name=mul_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=true_not_false
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=le_not_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=not_lt_le