Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro
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
j1 j2 : Fin acc.snd.fst.length
j1_eq_i : (acc.snd.fst.get j1).fst.val = ↑i
j2_eq_i : (acc.snd.fst.get j2).fst.val = ↑i
j1_eq_true : (acc.snd.fst.get j1).snd = true
j2_eq_false : (acc.snd.fst.get j2).snd = false
h1 : acc.fst[↑i] = both
h2 : f.assignments[↑i] = unassigned
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j1 → k ≠ j2 → (acc.snd.fst.get k).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:
. have j1_succ_in_bounds : j1.1 + 1 < (l :: acc.2.1).length := by
    simp only [List.length_cons, Nat.succ_eq_add_one]
    exact Nat.succ_lt_succ j1.2
  have j2_succ_in_bounds : j2.1 + 1 < (l :: acc.2.1).length := by
    simp only [List.length_cons, Nat.succ_eq_add_one]
    exact Nat.succ_lt_succ j2.2
  let j1_succ : Fin (l :: acc.2.1).length := ⟨j1.1 + 1, j1_succ_in_bounds⟩
  let j2_succ : Fin (l :: acc.2.1).length := ⟨j2.1 + 1, j2_succ_in_bounds⟩
  apply Or.inr ∘ Or.inr ∘ Exists.intro j1_succ ∘ Exists.intro j2_succ
  simp only [List.get, Nat.add_eq, Nat.add_zero, List.length_cons, ne_eq]
  apply And.intro j1_eq_i ∘ And.intro j2_eq_i ∘ And.intro j1_eq_true ∘ And.intro j2_eq_false
  have l_ne_i : l.1.1 ≠ i.1 := by
    intro l_eq_i
    simp only [hasAssignment, Bool.not_eq_true] at h
    split at h
    all_goals
      simp (config := {decide := true}) [getElem!, l_eq_i, i_in_bounds, h1] at h
  constructor
  . rw [Array.get_modify_unchanged l_in_bounds i_in_bounds _ l_ne_i]
    exact h1
  . constructor
    . exact h2
    . intro k k_ne_j1_succ k_ne_j2_succ
      have zero_in_bounds : 0 < (l :: acc.2.1).length := by
        simp only [List.length_cons]
        exact Nat.zero_lt_succ (List.length acc.snd.fst)
      by_cases k = ⟨0, zero_in_bounds⟩
      . next 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 k'] 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_j1 : ⟨k', k'_in_bounds⟩ ≠ j1 := by
          simp only [List.length_cons, k_eq_succ, ne_eq, Fin.mk.injEq, Nat.succ.injEq, j1_succ] at k_ne_j1_succ
          exact Fin.ne_of_val_ne k_ne_j1_succ
        have k'_ne_j2 : ⟨k', k'_in_bounds⟩ ≠ j2 := by
          simp only [List.length_cons, k_eq_succ, ne_eq, Fin.mk.injEq, Nat.succ.injEq, j2_succ] at k_ne_j2_succ
          exact Fin.ne_of_val_ne k_ne_j2_succ
        exact h3 ⟨k', k'_in_bounds⟩ k'_ne_j1 k'_ne_j2
Proof state after:
no goals