def internal_sorted_cons_ins

bend-mathlib@0.8.1.0 · sort.bend:62

import Base
import bend-mathlib@0.8.1.0/list.bend as MList
import bend-mathlib@0.8.1.0/perm.bend as MPerm

62def internal_sorted_cons_ins(~A: Data, ~le: A -> A -> Bool, ~le_total: @x: A -> @y: A -> {le(x, y) == False{} : Bool} -> {le(y, x) == True{} : Bool}, +t: List<&2, A>, +y: A, +x: A, hyx: {le(y, x) == True{} : Bool}, +h: MList.sorted_by(~A, ~le, y <> t)) -> MList.sorted_by(~A, ~le, y <> MPerm.insert_by(~A, ~le, x, t)):
63  match t:
64    case Nil{}:
65      internal_sorted_cons_cons_intro(~A, ~le, y, x, Nil{}, hyx, internal_sorted_single(~A, ~le, x))
66    case +z <> +u:
67      internal_cons_ins_step(~A, ~le, ~le_total, le(x, z), y, x, z, u, {==}, hyx, h, hzx => internal_sorted_cons_ins(~A, ~le, ~le_total, u, z, x, hzx, internal_sorted_cons_cons_elim_tail(~A, ~le, y, z, u, h)))