def update_append_right

bend-collections-laws-math@1.0.0.0 · proofs/lib/list.bend:90

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

90def update_append_right(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>, +i: Nat, +v: A, +h: {Nat.is_le(SC.length(A, xs), i) == True{} : Bool}) -> {SC.update(A, SC.append(A, xs, ys), i, v) == SC.append(A, xs, SC.update(A, ys, Nat.sub(i, SC.length(A, xs)), v)) : List<&2, A>}:
91  match xs i:
92    case Nil{} _:
93      %Equal.sym(Nat, Nat.sub(i, 0n), i, N.sub_zero(i)) : {SC.update(A, ys, i, v) == SC.update(A, ys, _, v) : List<&2, A>}
94      {==}
95    case Con{h1, t} 0n:
96      Empty.absurd({SC.update(A, SC.append(A, Con{h1, t}, ys), 0n, v) == SC.append(A, Con{h1, t}, SC.update(A, ys, Nat.sub(0n, 1n+SC.length(A, t)), v)) : List<&2, A>}, L.false_true(h))
97    case Con{+h1, t} 1n+p:
98      cons_cong(A, h1, SC.update(A, SC.append(A, t, ys), p, v), SC.append(A, t, SC.update(A, ys, Nat.sub(p, SC.length(A, t)), v)), update_append_right(A, t, ys, p, v, h))