def nth_append_right

bend-collections-laws-math@1.0.0.0 · proofs/lib/list.bend:71

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as SC

71def nth_append_right(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>, +i: Nat, +h: {Nat.is_le(SC.length(A, xs), i) == True{} : Bool}) -> {SC.nth(A, SC.append(A, xs, ys), i) == SC.nth(A, ys, Nat.sub(i, SC.length(A, xs))) : Maybe<&2, A>}:
72  match xs i:
73    case Nil{} _:
74      %Equal.sym(Nat, Nat.sub(i, 0n), i, N.sub_zero(i)) : {SC.nth(A, ys, i) == SC.nth(A, ys, _) : Maybe<&2, A>}
75      {==}
76    case Con{h1, t} 0n:
77      Empty.absurd({SC.nth(A, SC.append(A, Con{h1, t}, ys), 0n) == SC.nth(A, ys, Nat.sub(0n, 1n+SC.length(A, t))) : Maybe<&2, A>}, L.false_true(h))
78    case Con{h1, t} 1n+p:
79      nth_append_right(A, t, ys, p, h)