Reduce.fold_lt2_h: 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 106-119 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=106&end_line=119

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/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/semiring.bend as SR
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/ineq.bend as I

# (the hi >= 1 case) lo + h t < d with d + c <= t: h = 0, and then lo + c < t
def fold_lt2_h(+t: Nat, +lo: Nat, +cv: Nat, +d: Nat, +f: Nat, +hd: {Nat.is_le(Nat.add(d, cv), t) == True{} : Bool}, h: Nat, +hs: {Nat.is_lt(Nat.add(lo, Nat.mul(h, t)), d) == True{} : Bool}, +ef: {f == Nat.add(lo, Nat.mul(1n+h, cv)) : Nat}) -> {Nat.is_lt(f, t) == True{} : Bool}:
  match h:
    case 0n:
      +hl = I.lt_subst_l(Nat.add(lo, 0n), lo, d, NA.add_zero(lo), hs)
      +s1 = I.lt_le_add(lo, cv, d, cv, hl, N.le_refl(cv))
      +s2 = N.lt_le_trans(Nat.add(lo, cv), Nat.add(d, cv), t, s1, hd)
      %Equal.sym(Nat, f, Nat.add(lo, Nat.mul(1n, cv)), ef) : {Nat.is_lt(_, t) == True{} : Bool}
      %Equal.sym(Nat, Nat.mul(1n, cv), cv, SR.one_mul(cv)) : {Nat.is_lt(Nat.add(lo, _), t) == True{} : Bool}
      s2
    case 1n+ +g:
      +le1 = N.le_trans(d, Nat.add(d, cv), t, N.le_add_right(d, cv), hd)
      +le2 = N.le_trans(t, Nat.add(t, Nat.mul(g, t)), Nat.add(lo, Nat.add(t, Nat.mul(g, t))), N.le_add_right(t, Nat.mul(g, t)), I.le_add_l(lo, Nat.add(t, Nat.mul(g, t))))
      Empty.absurd({Nat.is_lt(f, t) == True{} : Bool}, Lg.true_not_false(Nat.is_lt(Nat.add(lo, Nat.add(t, Nat.mul(g, t))), d), hs, N.le_not_lt(Nat.add(lo, Nat.add(t, Nat.mul(g, t))), d, N.le_trans(d, t, Nat.add(lo, Nat.add(t, Nat.mul(g, t))), le1, le2))))

Uses (19):
  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=Equal.sym
  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=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/ineq.bend&name=le_add_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/ineq.bend&name=lt_le_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/ineq.bend&name=lt_subst_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/semiring.bend&name=one_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_zero
  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_add_right
  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=le_refl
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_le_trans