-- 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:
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
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
⊢ 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:
specialize hf i b hb

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 : 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

Proof state:
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
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
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✝² : Option (DefaultClause n)
val✝ : DefaultClause n
x✝¹ :
  ∀ (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✝
x✝ : Option (DefaultClause n)
heq2 : f.clauses[id]! = none
⊢ { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments } =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }

Tactic:
simp [heq] at heq2

Result:
no goals

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

Tactic:
by_cases hl : ∃ l : Literal (PosFin n), c = unit l

Result:
case pos
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
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl : ∃ l, c = unit l
⊢ unit (i, b) ∈
    (match some c with
      | none => { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
      | some { clause := [l], nodupkey := nodupkey, nodup := nodup } =>
        { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }
      | some val =>
        { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments }).toList

case neg
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
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl : ¬∃ l, c = unit l
⊢ unit (i, b) ∈
    (match some c with
      | none => { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
      | some { clause := [l], nodupkey := nodupkey, nodup := nodup } =>
        { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }
      | some val =>
        { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments }).toList

Proof state:
case inl.h
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
⊢ some (unit (i, b)) ∈ (f.clauses.set! id none).toList

Tactic:
simp only [Array.set!, Array.setIfInBounds]

Result:
case inl.h
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
⊢ some (unit (i, b)) ∈ (if h : id < f.clauses.size then f.clauses.set id none h else f.clauses).toList

Proof state:
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
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
heq : f.clauses[id]! = none
hb : hasAssignment b f.assignments[i.val] = true
⊢ unit (i, b) ∈
    { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }.toList

Tactic:
exact hf i b hb

Result:
no goals