-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
theorem reduce_postcondition {n : Nat} (c : DefaultClause n) (assignment : Array Assignment) :
    (reduce c assignment = reducedToEmpty → Incompatible (PosFin n) c assignment) ∧
    (∀ l : Literal (PosFin n), reduce c assignment = reducedToUnit l → ∀ (p : (PosFin n) → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l)

Proof state:
case intro.left.inr
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
h_inductive :
  ∀ (idx : Fin c_arr.size) (res : ReduceResult (PosFin n)),
    motive (↑idx) res → motive (↑idx + 1) (reduce_fold_fn assignment res c_arr[idx])
h2 :
  ∀ (l : Literal (PosFin n)),
    Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
      ∀ (p : PosFin n → Bool), p ⊨ assignment → (∃ i, ↑i < c_arr.size ∧ (p ⊨ c_arr[i])) → p ⊨ l
h : Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty
p : PosFin n → Bool
h1 : ¬(p ⊨ assignment)
⊢ ¬(p ⊨ c) ∨ ¬(p ⊨ assignment)

Tactic:
exact Or.inr h1

Result:
no goals

Proof state:
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
⊢ (Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty →
      Incompatible (PosFin n) c assignment) ∧
    ∀ (l : Literal (PosFin n)),
      Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
        ∀ (p : PosFin n → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l

Tactic:
have h_base : motive 0 reducedToEmpty := by
  have : ∀ (a : PosFin n) (b : Bool), (reducedToEmpty = reducedToUnit (a, b)) = False := by intros; simp
  simp only [ReducePostconditionInductionMotive, Fin.getElem_fin, forall_exists_index, and_imp, Prod.forall,
    forall_const, false_implies, implies_true, and_true, motive, this]
  intro p
  apply Or.inl
  intro i i_lt_zero
  exfalso
  exact Nat.not_lt_zero i.1 i_lt_zero

Result:
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
⊢ (Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty →
      Incompatible (PosFin n) c assignment) ∧
    ∀ (l : Literal (PosFin n)),
      Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
        ∀ (p : PosFin n → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l

Proof state:
case intro
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
h_inductive :
  ∀ (idx : Fin c_arr.size) (res : ReduceResult (PosFin n)),
    motive (↑idx) res → motive (↑idx + 1) (reduce_fold_fn assignment res c_arr[idx])
h1 :
  Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty →
    ∀ (p : PosFin n → Bool), (∀ (i : Fin c_arr.size), ↑i < c_arr.size → ¬(p ⊨ c_arr[i])) ∨ ¬(p ⊨ assignment)
h2 :
  ∀ (l : Literal (PosFin n)),
    Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
      ∀ (p : PosFin n → Bool), p ⊨ assignment → (∃ i, ↑i < c_arr.size ∧ (p ⊨ c_arr[i])) → p ⊨ l
⊢ (Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty →
      Incompatible (PosFin n) c assignment) ∧
    ∀ (l : Literal (PosFin n)),
      Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
        ∀ (p : PosFin n → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l

Tactic:
constructor

Result:
case intro.left
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
h_inductive :
  ∀ (idx : Fin c_arr.size) (res : ReduceResult (PosFin n)),
    motive (↑idx) res → motive (↑idx + 1) (reduce_fold_fn assignment res c_arr[idx])
h1 :
  Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty →
    ∀ (p : PosFin n → Bool), (∀ (i : Fin c_arr.size), ↑i < c_arr.size → ¬(p ⊨ c_arr[i])) ∨ ¬(p ⊨ assignment)
h2 :
  ∀ (l : Literal (PosFin n)),
    Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
      ∀ (p : PosFin n → Bool), p ⊨ assignment → (∃ i, ↑i < c_arr.size ∧ (p ⊨ c_arr[i])) → p ⊨ l
⊢ Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty → Incompatible (PosFin n) c assignment

case intro.right
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
h_inductive :
  ∀ (idx : Fin c_arr.size) (res : ReduceResult (PosFin n)),
    motive (↑idx) res → motive (↑idx + 1) (reduce_fold_fn assignment res c_arr[idx])
h1 :
  Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToEmpty →
    ∀ (p : PosFin n → Bool), (∀ (i : Fin c_arr.size), ↑i < c_arr.size → ¬(p ⊨ c_arr[i])) ∨ ¬(p ⊨ assignment)
h2 :
  ∀ (l : Literal (PosFin n)),
    Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
      ∀ (p : PosFin n → Bool), p ⊨ assignment → (∃ i, ↑i < c_arr.size ∧ (p ⊨ c_arr[i])) → p ⊨ l
⊢ ∀ (l : Literal (PosFin n)),
    Array.foldl (reduce_fold_fn assignment) reducedToEmpty c_arr = reducedToUnit l →
      ∀ (p : PosFin n → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l