internlm/Lean-Github huggingface dataset:
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inl.intro.inl
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, false) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, false) = c
h_insertUnit_fold : (i, false) ∈ f.rupUnits.data
⊢ (i, false) ∈ f.rupUnits.data ∧ unit (i, false) = c
Tactic:
exact ⟨h_insertUnit_fold, h2⟩
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inl.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, false) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, false) = c
h_insertUnit_fold : (i, false) ∈ units
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
apply Or.inl ∘ Exists.intro i ∘ Or.inl
Proof state after:
case inr.inl.intro.inl.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, false) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, false) = c
h_insertUnit_fold : (i, false) ∈ units
⊢ (i, false) ∈ units ∧ Clause.unit (i, false) = c
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inl.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, false) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, false) = c
h_insertUnit_fold : (i, false) ∈ units
⊢ (i, false) ∈ units ∧ Clause.unit (i, false) = c
Tactic:
exact ⟨h_insertUnit_fold, h2⟩
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inr.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ units
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
. apply Or.inl ∘ Exists.intro i ∘ Or.inr
  exact ⟨h_insertUnit_fold, h2⟩
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inr.intro.inl
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ f.rupUnits.data
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
apply Or.inr ∘ Or.inr ∘ Or.inl ∘ Exists.intro i ∘ Or.inr
Proof state after:
case inr.inl.intro.inr.intro.inl
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ f.rupUnits.data
⊢ (i, true) ∈ f.rupUnits.data ∧ unit (i, true) = c
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inr.intro.inl
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ f.rupUnits.data
⊢ (i, true) ∈ f.rupUnits.data ∧ unit (i, true) = c
Tactic:
exact ⟨h_insertUnit_fold, h2⟩
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inr.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ units
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
apply Or.inl ∘ Exists.intro i ∘ Or.inr
Proof state after:
case inr.inl.intro.inr.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ units
⊢ (i, true) ∈ units ∧ Clause.unit (i, true) = c
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inl.intro.inr.intro.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold✝ :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
i : PosFin n
h1 : (i, true) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data
h2 : unit (i, true) = c
h_insertUnit_fold : (i, true) ∈ units
⊢ (i, true) ∈ units ∧ Clause.unit (i, true) = c
Tactic:
exact ⟨h_insertUnit_fold, h2⟩
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.mem_of_insertRupUnits
Proof state before:
case inr.inr
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.rupUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h_insertUnit_fold :
  ∀ (l : Literal (PosFin n)),
    l ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.data → l ∈ f.rupUnits.data ∨ l ∈ units
h : ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
exact (Or.inr ∘ Or.inr ∘ Or.inr) h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.mem_of_insertRatUnits
Proof state before:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
⊢ c ∈ (f.insertRatUnits units).fst.toList → c ∈ List.map Clause.unit units ∨ c ∈ f.toList
Tactic:
simp only [toList, insertRatUnits, Prod.mk.eta, Array.toList_eq, List.append_assoc, List.mem_append,
  List.mem_filterMap, id_eq, exists_eq_right, List.mem_map, Prod.exists, Bool.exists_bool]
Proof state after:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
⊢ (some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a,
          (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
            (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧
              unit (a, true) = c) →
    (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
      some c ∈ f.clauses.data ∨
        (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
          ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Theorem: LRAT.DefaultFormula.mem_of_insertRatUnits
Proof state before:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
⊢ (some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a,
          (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
            (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧
              unit (a, true) = c) →
    (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
      some c ∈ f.clauses.data ∨
        (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
          ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
intro h
Proof state after:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
h :
  some c ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
      ∃ a,
        (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
          (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, true) = c
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Theorem: LRAT.DefaultFormula.mem_of_insertRatUnits
Proof state before:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
h :
  some c ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
      ∃ a,
        (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
          (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, true) = c
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
have hb : ∀ l : Literal (PosFin n), l ∈ (f.ratUnits, f.assignments, false).1.data → (l ∈ f.ratUnits.data ∨ l ∈ units) :=
  fun _ => Or.inl
Proof state after:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
h :
  some c ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
      ∃ a,
        (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
          (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, true) = c
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.ratUnits, f.assignments, false).fst.data → l ∈ f.ratUnits.data ∨ l ∈ units
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Theorem: LRAT.DefaultFormula.mem_of_insertRatUnits
Proof state before:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
h :
  some c ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
      ∃ a,
        (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
          (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, true) = c
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.ratUnits, f.assignments, false).fst.data → l ∈ f.ratUnits.data ∨ l ∈ units
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c
Tactic:
have hl (acc : Array (Literal (PosFin n)) × Array Assignment × Bool)
  (ih : ∀ l : Literal (PosFin n), l ∈ acc.1.data → l ∈ f.ratUnits.data ∨ l ∈ units)
  (unit : Literal (PosFin n)) (unit_in_units : unit ∈ units) :
  ∀ l : Literal (PosFin n), l ∈ (insertUnit acc unit).1.data → (l ∈ f.ratUnits.data ∨ l ∈ units) := by
  intro l hl
  rw [insertUnit] at hl
  simp only [Prod.mk.eta] at hl
  split at hl
  . exact ih l hl
  . simp only [Array.push_data, List.mem_append, List.mem_singleton] at hl
    rcases hl with l_in_acc | l_eq_unit
    . exact ih l l_in_acc
    . rw [l_eq_unit]
      exact Or.inr unit_in_units
Proof state after:
n : Nat
f : DefaultFormula n
units : List (Literal (PosFin n))
c : DefaultClause n
h :
  some c ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
      ∃ a,
        (a, false) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, false) = c ∨
          (a, true) ∈ (List.foldl insertUnit (f.ratUnits, f.assignments, false) units).fst.data ∧ unit (a, true) = c
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.ratUnits, f.assignments, false).fst.data → l ∈ f.ratUnits.data ∨ l ∈ units
hl :
  ∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
    (∀ (l : Literal (PosFin n)), l ∈ acc.fst.data → l ∈ f.ratUnits.data ∨ l ∈ units) →
      ∀ (unit : Literal (PosFin n)),
        unit ∈ units → ∀ (l : Literal (PosFin n)), l ∈ (insertUnit acc unit).fst.data → l ∈ f.ratUnits.data ∨ l ∈ units
⊢ (∃ a, (a, false) ∈ units ∧ Clause.unit (a, false) = c ∨ (a, true) ∈ units ∧ Clause.unit (a, true) = c) ∨
    some c ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c