PROOFOrder.wd.fin_trans: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_order.bend as PROOFOrder
Lines 247-267 of bend-osdev-u32@0.1.0.0/PROOF_order.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&start_line=247&end_line=267

import Base

def wd.fin_trans(c1: Cmp, c2: Cmp, +p: Nat, +x: Bool, +y: Bool, +z: Bool, +at: Word(p), +bt: Word(p), +ct: Word(p), +e1: {Word.cmp(p, at, bt) == c1 : Cmp}, +e2: {Word.cmp(p, bt, ct) == c2 : Cmp}, +h1: {Word.cmp.fin(x, y, c1) == LT{} : Cmp}, +h2: {Word.cmp.fin(y, z, c2) == LT{} : Cmp}, ih: {Word.cmp(p, at, bt) == LT{} : Cmp} -> {Word.cmp(p, bt, ct) == LT{} : Cmp} -> {Word.cmp(p, at, ct) == LT{} : Cmp}) -> {Word.cmp.fin(x, z, Word.cmp(p, at, ct)) == LT{} : Cmp}:
  match c1 c2:
    case LT{} LT{}:
      %Equal.sym(Cmp, Word.cmp(p, at, ct), LT{}, ih(e1, e2)) : {Word.cmp.fin(x, z, _) == LT{} : Cmp}
      {==}
    case LT{} EQ{}:
      %wd.eq(p, bt, ct, e2) : {Word.cmp.fin(x, z, Word.cmp(p, at, _)) == LT{} : Cmp}
      %Equal.sym(Cmp, Word.cmp(p, at, bt), LT{}, e1) : {Word.cmp.fin(x, z, _) == LT{} : Cmp}
      {==}
    case EQ{} LT{}:
      %Equal.sym(Word(p), at, bt, wd.eq(p, at, bt, e1)) : {Word.cmp.fin(x, z, Word.cmp(p, _, ct)) == LT{} : Cmp}
      %Equal.sym(Cmp, Word.cmp(p, bt, ct), LT{}, e2) : {Word.cmp.fin(x, z, _) == LT{} : Cmp}
      {==}
    case EQ{} EQ{}:
      %Equal.trans(Word(p), at, bt, ct, wd.eq(p, at, bt, e1), wd.eq(p, bt, ct, e2)) : {Word.cmp.fin(x, z, Word.cmp(p, at, _)) == LT{} : Cmp}
      %Equal.sym(Cmp, Word.cmp(p, at, at), EQ{}, wd.refl(p, at)) : {Word.cmp.fin(x, z, _) == LT{} : Cmp}
      wd.bool_trans(x, y, z, h1, h2)
    case GT{} c2:
      Empty.absurd({Word.cmp.fin(x, z, Word.cmp(p, at, ct)) == LT{} : Cmp}, ab.gt_lt(h1))
    case c1 GT{}:
      Empty.absurd({Word.cmp.fin(x, z, Word.cmp(p, at, ct)) == LT{} : Cmp}, ab.gt_lt(h2))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Cmp
  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=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.cmp
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.cmp.fin
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&name=ab.gt_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&name=wd.bool_trans
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&name=wd.eq
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&name=wd.refl