-- 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.inr.intro.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
j1 j2 : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j1 → k ≠ j2 → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
i_true_in_insertUnit_fold : (i, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.toList
i_false_in_insertUnit_fold : (i, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.toList
⊢ False

Tactic:
simp only [hf.1, negate, Literal.negate] at i_true_in_insertUnit_fold i_false_in_insertUnit_fold

Result:
case intro.inr.inr.intro.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
j1 j2 : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j1 → k ≠ j2 → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
i_true_in_insertUnit_fold :
  (i, true) ∈ (List.foldl insertUnit (#[], f.assignments, false) (List.map Literal.negate c.clause)).fst.toList
i_false_in_insertUnit_fold :
  (i, false) ∈ (List.foldl insertUnit (#[], f.assignments, false) (List.map Literal.negate c.clause)).fst.toList
⊢ False

Proof state:
case intro.inr.inr.intro.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
j1 j2 : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j1 → k ≠ j2 → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
i_true_in_insertUnit_fold✝ :
  (i, true) ∈ (List.foldl insertUnit (#[], f.assignments, false) (List.map Literal.negate c.clause)).fst.toList
i_false_in_insertUnit_fold✝ :
  (i, false) ∈ (List.foldl insertUnit (#[], f.assignments, false) (List.map Literal.negate c.clause)).fst.toList
i_true_in_insertUnit_fold : (i, false) ∈ c.clause
i_false_in_insertUnit_fold : (i, true) ∈ c.clause
c_not_tautology : ¬(i, true) ∈ Clause.toList c ∨ ¬Literal.negate (i, true) ∈ Clause.toList c
⊢ False

Tactic:
simp only [Clause.toList] at c_not_tautology

Result:
case intro.inr.inr.intro.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
j1 j2 : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
  ∀ (k : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
    k ≠ j1 → k ≠ j2 → (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
i_true_in_insertUnit_fold✝ :
  (i, true) ∈ (List.foldl insertUnit (#[], f.assignments, false) (List.map Literal.negate c.clause)).fst.toList
i_false_in_insertUnit_fold✝ :
  (i, false) ∈ (List.foldl insertUnit (#[], f.assignments, false) (List.map Literal.negate c.clause)).fst.toList
i_true_in_insertUnit_fold : (i, false) ∈ c.clause
i_false_in_insertUnit_fold : (i, true) ∈ c.clause
c_not_tautology : ¬(i, true) ∈ c.toList ∨ ¬Literal.negate (i, true) ∈ c.toList
⊢ False