-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean
theorem readyForRupAdd_insert {n : Nat} (f : DefaultFormula n) (c : DefaultClause n) :
    ReadyForRupAdd f → ReadyForRupAdd (insert f c)

Proof state:
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
ib_ne_c : ¬(i, b) = (l, true)
l_eq_i : l.val = i.val
⊢ b = false

Tactic:
by_cases b = true

Result:
case pos
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
ib_ne_c : ¬(i, b) = (l, true)
l_eq_i : l.val = i.val
h✝ : b = true
⊢ b = false

case neg
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
ib_ne_c : ¬(i, b) = (l, true)
l_eq_i : l.val = i.val
h✝ : ¬b = true
⊢ b = false

Proof state:
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
ib_ne_c : ¬(i, b) = (l, true)
l_ne_i : ¬l.val = i.val
⊢ hasAssignment b f.assignments[i.val] = true

Tactic:
simp only [Array.getElem_modify_of_ne l_ne_i] at hb

Result:
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
ib_ne_c : ¬(i, b) = (l, true)
l_ne_i : ¬l.val = i.val
hb : hasAssignment b f.assignments[i.val] = true
⊢ hasAssignment b f.assignments[i.val] = true

Proof state:
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, false)
hsize : (f.assignments.modify l.val addNegAssignment).size = n
i : PosFin n
b : Bool
hb :
  hasAssignment b
      { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addNegAssignment }.assignments[i.val] =
    true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
⊢ unit (i, b) ∈
    { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.val addNegAssignment }.toList

Tactic:
have i_in_bounds : i.1 < f.assignments.size := by rw [f_readyForRupAdd.2.1]; exact i.2.2

Result:
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, false)
hsize : (f.assignments.modify l.val addNegAssignment).size = n
i : PosFin n
b : Bool
hb :
  hasAssignment b
      { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addNegAssignment }.assignments[i.val] =
    true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
⊢ unit (i, b) ∈
    { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.val addNegAssignment }.toList

Proof state:
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
⊢ unit (i, b) ∈
    { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.val addPosAssignment }.toList

Tactic:
by_cases (i, b) = (l, true)

Result:
case pos
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
h✝ : (i, b) = (l, true)
⊢ unit (i, b) ∈
    { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.val addPosAssignment }.toList

case neg
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, true)
hsize : (f.assignments.modify l.val addPosAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addPosAssignment)[i.val] = true
hf : hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
i_in_bounds : i.val < f.assignments.size
h✝ : ¬(i, b) = (l, true)
⊢ unit (i, b) ∈
    { clauses := f.clauses.push (some c), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.val addPosAssignment }.toList