Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case neg
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
h✝ : ¬l.fst.val = ↑i
⊢ let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i := (acc.fst.modify l.fst.val (addAssignment l.snd))[↑i];
  let fassignments_i := f.assignments[↑i];
  (assignments_i = fassignments_i ∧ ∀ (l_1 : Literal (PosFin n)), l_1 ∈ l :: acc.snd.fst → l_1.fst.val ≠ ↑i) ∨
    (∃ j,
        ((l :: acc.snd.fst).get j).fst.val = ↑i ∧
          assignments_i = addAssignment ((l :: acc.snd.fst).get j).snd fassignments_i ∧
            ¬hasAssignment ((l :: acc.snd.fst).get j).snd fassignments_i = true ∧
              ∀ (k : Fin (l :: acc.snd.fst).length), k ≠ j → ((l :: acc.snd.fst).get k).fst.val ≠ ↑i) ∨
      ∃ j1 j2,
        ((l :: acc.snd.fst).get j1).fst.val = ↑i ∧
          ((l :: acc.snd.fst).get j2).fst.val = ↑i ∧
            ((l :: acc.snd.fst).get j1).snd = true ∧
              ((l :: acc.snd.fst).get j2).snd = false ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin (l :: acc.snd.fst).length), k ≠ j1 → k ≠ j2 → ((l :: acc.snd.fst).get k).fst.val ≠ ↑i
Tactic:
next l_ne_i =>
apply Or.inr ∘ Or.inl ∘ Exists.intro ⟨j.1 + 1, j_succ_in_bounds⟩
simp only [List.get, Nat.add_eq, Nat.add_zero]
constructor
. exact j_eq_i
. constructor
  . rw [Array.get_modify_unchanged l_in_bounds i_in_bounds _ l_ne_i]
    exact h1
  . apply And.intro h2
    intro k k_ne_j_succ
    by_cases k.1 = 0
    . next k_eq_zero =>
      have k_eq_zero : k = ⟨0, zero_in_bounds⟩ := by
        apply Fin.eq_of_val_eq
        simp only [List.length_cons]
        exact k_eq_zero
      simp only [k_eq_zero, List.length_cons, List.get, ne_eq]
      exact l_ne_i
    . next k_ne_zero =>
      have k_eq_succ : ∃ k' : Nat, ∃ k'_succ_in_bounds : k' + 1 < (l :: acc.2.1).length, k = ⟨k' + 1, k'_succ_in_bounds⟩ := by
        have k_val_ne_zero : k.1 ≠ 0 := by
          intro k_eq_zero
          simp only [List.length_cons, ← k_eq_zero, ne_eq, not_true] at k_ne_zero
        rcases Nat.exists_eq_succ_of_ne_zero k_val_ne_zero with ⟨k', k_eq_k'_succ⟩
        rw [Nat.succ_eq_add_one] at k_eq_k'_succ
        have k'_succ_in_bounds : k' + 1 < (l :: acc.2.1).length := by rw [← k_eq_k'_succ]; exact k.2
        apply Exists.intro k' ∘ Exists.intro k'_succ_in_bounds
        apply Fin.eq_of_val_eq
        exact k_eq_k'_succ
      rcases k_eq_succ with ⟨k', k'_succ_in_bounds, k_eq_succ⟩
      rw [k_eq_succ]
      simp only [List.get, Nat.add_eq, Nat.add_zero, ne_eq]
      have k'_in_bounds : k' < acc.2.1.length := by
        simp only [List.length_cons, Nat.succ_eq_add_one] at k'_succ_in_bounds
        exact Nat.lt_of_succ_lt_succ k'_succ_in_bounds
      have k'_ne_j : ⟨k', k'_in_bounds⟩ ≠ j := by
        simp only [List.length_cons] at k_eq_succ
        simp only [List.length_cons, k_eq_succ, ne_eq, Fin.mk.injEq, Nat.succ.injEq] at k_ne_j_succ
        exact Fin.ne_of_val_ne k_ne_j_succ
      exact h3 ⟨k', k'_in_bounds⟩ k'_ne_j
Proof state after:
no goals