-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean
theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
    StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id)

Proof state:
case neg.h_3.inl.h.isTrue.intro.intro
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
  ∀ (x : PosFin n),
    ¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
      ¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
  ∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
    (nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
deleteOne_f_rw :
  f.deleteOne id =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
hb : hasAssignment b f.assignments[i.val] = true
hf : some (unit (i, b)) ∈ f.clauses.toList
h✝ : id < f.clauses.size
idx : Nat
hbound : idx < f.clauses.toList.length
hidx : f.clauses.toList[idx] = some (unit (i, b))
⊢ ∃ n_1, (f.clauses.toList.set id none).get n_1 = f.clauses.toList[idx]

Tactic:
have idx_in_bounds : idx < List.length (List.set f.clauses.toList id none) := by
  simp only [List.length_set]
  exact hbound

Result:
case neg.h_3.inl.h.isTrue.intro.intro
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
  ∀ (x : PosFin n),
    ¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
      ¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
  ∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
    (nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
deleteOne_f_rw :
  f.deleteOne id =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
hb : hasAssignment b f.assignments[i.val] = true
hf : some (unit (i, b)) ∈ f.clauses.toList
h✝ : id < f.clauses.size
idx : Nat
hbound : idx < f.clauses.toList.length
hidx : f.clauses.toList[idx] = some (unit (i, b))
idx_in_bounds : idx < (f.clauses.toList.set id none).length
⊢ ∃ n_1, (f.clauses.toList.set id none).get n_1 = f.clauses.toList[idx]

Proof state:
n : Nat
f : DefaultFormula n
id : Nat
hf : f.StrongAssignmentsInvariant
⊢ (f.deleteOne id).StrongAssignmentsInvariant

Tactic:
rcases hf with ⟨hsize, hf⟩

Result:
case intro
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
⊢ (f.deleteOne id).StrongAssignmentsInvariant

Proof state:
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_ne_i : ¬l.fst.val = i.val
hb : hasAssignment b f.assignments[i.val] = true
hf : some (unit (i, b)) ∈ f.clauses.toList
h✝ : id < f.clauses.size
idx : Nat
hbound : idx < f.clauses.toList.length
hidx : f.clauses.toList[idx] = some (unit (i, b))
⊢ idx < f.clauses.toList.length

Tactic:
exact hbound

Result:
no goals

Proof state:
case pos
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
  ∀ (x : PosFin n),
    ¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
      ¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
  ∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
    (nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
deleteOne_f_rw :
  f.deleteOne id =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
hb : hasAssignment b f.assignments[i.val] = true
hf : some (unit (i, b)) ∈ f.clauses.toList
h✝¹ : id < f.clauses.size
idx : Nat
hbound : idx < f.clauses.toList.length
hidx : f.clauses.toList[idx] = some (unit (i, b))
idx_in_bounds : idx < (f.clauses.toList.set id none).length
h✝ : id = idx
⊢ (f.clauses.toList.set id none).get ⟨idx, idx_in_bounds⟩ = f.clauses.toList[idx]

Tactic:
next id_eq_idx =>
exfalso
have idx_in_bounds2 : idx < f.clauses.size := by
  conv => rhs; rw [Array.size_toArray]
  exact hbound
simp only [getElem!, id_eq_idx, Array.length_toList, idx_in_bounds2, ↓reduceDIte,
  Fin.eta, Array.get_eq_getElem, ← Array.getElem_toList, decidableGetElem?] at heq
rw [hidx] at heq
simp only [Option.some.injEq] at heq
rw [← heq] at hl
specialize hl i
simp only [unit, DefaultClause.mk.injEq, List.cons.injEq, Prod.mk.injEq, true_and, and_true,
  Bool.not_eq_false, Bool.not_eq_true] at hl
by_cases b_val : b <;> simp [b_val] at hl

Result:
no goals

Proof state:
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_eq_i : l.fst.val = i.val
l_ne_b : l.snd = !b
hb✝ : hasAssignment b (removeAssignment (!b) f.assignments[i.val]) = true
hb : hasAssignment b f.assignments[i.val] = true
hf : unit (i, b) ∈ f.toList
⊢ unit (i, b) ∈
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList

Tactic:
simp only [toList, List.append_assoc, List.mem_append, List.mem_filterMap, id_eq,
  exists_eq_right, List.mem_map, Prod.exists, Bool.exists_bool] at hf

Result:
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_eq_i : l.fst.val = i.val
l_ne_b : l.snd = !b
hb✝ : hasAssignment b (removeAssignment (!b) f.assignments[i.val]) = true
hb : hasAssignment b f.assignments[i.val] = true
hf :
  some (unit (i, b)) ∈ f.clauses.toList ∨
    (∃ a,
        (a, false) ∈ f.rupUnits.toList ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.rupUnits.toList ∧ unit (a, true) = unit (i, b)) ∨
      ∃ a,
        (a, false) ∈ f.ratUnits.toList ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.ratUnits.toList ∧ unit (a, true) = unit (i, b)
⊢ unit (i, b) ∈
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList