-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean
theorem sat_of_insertRat {n : Nat} (f : DefaultFormula n)
    (hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (p : PosFin n → Bool)
    (pf : p ⊨ f) :
    (insertRatUnits f (negate c)).2 = true → p ⊨ c

Proof state:
case intro.inr.inl.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.ratUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
  let update_res := List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate;
  let_fun update_res_size := ⋯;
  InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
j : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
b : Bool
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b)
h2 :
  (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] =
    addAssignment b f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
⊢ ∃ a, (a, false) ∈ Clause.toList c ∧ decide (p a = false) = true ∨ (a, true) ∈ Clause.toList c ∧ p a = true

Tactic:
apply Exists.intro i

Result:
case intro.inr.inl.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.ratUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
  let update_res := List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate;
  let_fun update_res_size := ⋯;
  InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
j : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
b : Bool
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b)
h2 :
  (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] =
    addAssignment b f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
⊢ (i, false) ∈ Clause.toList c ∧ decide (p i = false) = true ∨ (i, true) ∈ Clause.toList c ∧ p i = true

Proof state:
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
⊢ i.val < f.assignments.size

Tactic:
rw [hf.2.1]

Result:
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
⊢ i.val < n

Proof state:
case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inl.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.ratUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
  let update_res := List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate;
  let_fun update_res_size := ⋯;
  InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
j : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
b : Bool
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b)
h2 : both = addAssignment b f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
ib_in_insertUnit_fold : (i, b) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.toList
i' : PosFin n
i_false_in_c : (i, false) ∈ c.clause
i'_eq_i : i' = i
b_eq_true : true = b
⊢ decide (p i = false) = true

Tactic:
simp only [addAssignment, ← b_eq_true, addPosAssignment, ite_true] at h2

Result:
case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inl.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.ratUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
  let update_res := List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate;
  let_fun update_res_size := ⋯;
  InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
j : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
b : Bool
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b)
h3 : ¬hasAssignment b f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
ib_in_insertUnit_fold : (i, b) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.toList
i' : PosFin n
i_false_in_c : (i, false) ∈ c.clause
i'_eq_i : i' = i
b_eq_true : true = b
h2 :
  both =
    match f.assignments[i.val] with
    | pos => pos
    | neg => both
    | both => both
    | unassigned => pos
⊢ decide (p i = false) = true

Proof state:
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
⊢ (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true → p ⊨ c

Tactic:
intro insertUnit_fold_success

Result:
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
⊢ p ⊨ c