internlm/Lean-Github huggingface dataset:
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ (f.insert c1).clauses.data
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2

case mp.h.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2

case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
. apply Or.inl
  simp only [insert] at h
  split at h
  all_goals
    simp only [Array.push_data, List.mem_append, List.mem_singleton, Option.some.injEq] at h
    rcases h with h | h
    . exact h
    . exact False.elim $ c2_ne_c1 h
Proof state after:
case mp.h.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2

case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2

case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
. rw [insert_preserves_rupUnits] at h
  exact Or.inr $ Or.inl h
Proof state after:
case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
. rw [insert_preserves_ratUnits] at h
  exact Or.inr $ Or.inr h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ (f.insert c1).clauses.data
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
apply Or.inl
Proof state after:
case mp.h.inl.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ (f.insert c1).clauses.data
⊢ some c2 ∈ f.clauses.data
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ (f.insert c1).clauses.data
⊢ some c2 ∈ f.clauses.data
Tactic:
simp only [insert] at h
Proof state after:
case mp.h.inl.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  some c2 ∈
    (match c1.isUnit with
        | none =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments }
        | some (l, true) =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addPosAssignment }
        | some (l, false) =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addNegAssignment }).clauses.data
⊢ some c2 ∈ f.clauses.data
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  some c2 ∈
    (match c1.isUnit with
        | none =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments }
        | some (l, true) =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addPosAssignment }
        | some (l, false) =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addNegAssignment }).clauses.data
⊢ some c2 ∈ f.clauses.data
Tactic:
split at h
Proof state after:
case mp.h.inl.h.h_1
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
heq✝ : c1.isUnit = none
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments }.clauses.data
⊢ some c2 ∈ f.clauses.data

case mp.h.inl.h.h_2
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, true)
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l✝.val addPosAssignment }.clauses.data
⊢ some c2 ∈ f.clauses.data

case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l✝.val addNegAssignment }.clauses.data
⊢ some c2 ∈ f.clauses.data
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_1
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
heq✝ : c1.isUnit = none
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments }.clauses.data
⊢ some c2 ∈ f.clauses.data

case mp.h.inl.h.h_2
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, true)
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l✝.val addPosAssignment }.clauses.data
⊢ some c2 ∈ f.clauses.data

case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l✝.val addNegAssignment }.clauses.data
⊢ some c2 ∈ f.clauses.data
Tactic:
all_goals
  simp only [Array.push_data, List.mem_append, List.mem_singleton, Option.some.injEq] at h
  rcases h with h | h
  . exact h
  . exact False.elim $ c2_ne_c1 h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l✝.val addNegAssignment }.clauses.data
⊢ some c2 ∈ f.clauses.data
Tactic:
simp only [Array.push_data, List.mem_append, List.mem_singleton, Option.some.injEq] at h
Proof state after:
case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : some c2 ∈ f.clauses.data ∨ c2 = c1
⊢ some c2 ∈ f.clauses.data
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : some c2 ∈ f.clauses.data ∨ c2 = c1
⊢ some c2 ∈ f.clauses.data
Tactic:
rcases h with h | h
Proof state after:
case mp.h.inl.h.h_3.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : some c2 ∈ f.clauses.data
⊢ some c2 ∈ f.clauses.data

case mp.h.inl.h.h_3.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : c2 = c1
⊢ some c2 ∈ f.clauses.data
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_3.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : some c2 ∈ f.clauses.data
⊢ some c2 ∈ f.clauses.data

case mp.h.inl.h.h_3.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : c2 = c1
⊢ some c2 ∈ f.clauses.data
Tactic:
. exact h
Proof state after:
case mp.h.inl.h.h_3.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : c2 = c1
⊢ some c2 ∈ f.clauses.data
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_3.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : c2 = c1
⊢ some c2 ∈ f.clauses.data
Tactic:
. exact False.elim $ c2_ne_c1 h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_3.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : some c2 ∈ f.clauses.data
⊢ some c2 ∈ f.clauses.data
Tactic:
exact h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inl.h.h_3.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : c2 = c1
⊢ some c2 ∈ f.clauses.data
Tactic:
exact False.elim $ c2_ne_c1 h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
rw [insert_preserves_rupUnits] at h
Proof state after:
case mp.h.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
exact Or.inr $ Or.inl h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  ∃ a,
    (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
      (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
rw [insert_preserves_ratUnits] at h
Proof state after:
case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mp.h.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ f.clauses.data ∨
    (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
Tactic:
exact Or.inr $ Or.inr h
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mpr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
⊢ (c2 = c1 ∨
      some c2 ∈ f.clauses.data ∨
        (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
          ∃ a,
            (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2) →
    some c2 ∈ (f.insert c1).clauses.data ∨
      (∃ a,
          (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
            (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
        ∃ a,
          (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
            (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
Tactic:
intro h
Proof state after:
case mpr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  c2 = c1 ∨
    some c2 ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mpr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h :
  c2 = c1 ∨
    some c2 ∈ f.clauses.data ∨
      (∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2) ∨
        ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
Tactic:
rcases h with h | h | h | h
Proof state after:
case mpr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : c2 = c1
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2

case mpr.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ f.clauses.data
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2

case mpr.inr.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2

case mpr.inr.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
Theorem: LRAT.DefaultFormula.insert_iff
Proof state before:
case mpr.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ f.clauses.data
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2

case mpr.inr.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2

case mpr.inr.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2
Tactic:
. apply Or.inl
  simp only [insert]
  split
  all_goals
    simp only [Array.push_data, List.mem_append, List.mem_singleton, Option.some.injEq]
    exact Or.inl h
Proof state after:
case mpr.inr.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2

case mpr.inr.inr.inr
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : ∃ a, (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.data ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.data ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.data ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.data ∧ unit (a, true) = c2