Theorem: LRAT.DefaultFormula.ratAdd_sound
Proof state before:
n : Nat
f : DefaultFormula n
c : DefaultClause n
pivot : Literal (PosFin n)
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
f' : DefaultFormula n
f_readyForRatAdd : f.readyForRatAdd
pivot_in_c : pivot ∈ Clause.toList c
ratAddSuccess :
  (if f.ratHintsExhaustive pivot ratHints = true then
      let negC := c.negate;
      match f.insertRupUnits negC with
      | (f, contradictionFound) =>
        if contradictionFound = true then (f, false)
        else
          match f.performRupCheck rupHints with
          | (f, derivedLits, derivedEmpty, encounteredError) =>
            if encounteredError = true then (f, false)
            else
              if derivedEmpty = true then (f, false)
              else
                let fold_fn := fun x ratHint =>
                  match x with
                  | (f, allChecksPassed) =>
                    if allChecksPassed = true then f.performRatCheck pivot.negateLiteral ratHint else (f, false);
                match Array.foldl fold_fn (f, true) ratHints 0 with
                | (f, allChecksPassed) =>
                  if (!allChecksPassed) = true then (f, false)
                  else
                    match f with
                    | { clauses := clauses, rupUnits := rupUnits, ratUnits := ratUnits, assignments := assignments } =>
                      let assignments := restoreAssignments assignments derivedLits;
                      let f :=
                        { clauses := clauses, rupUnits := rupUnits, ratUnits := ratUnits,
                            assignments := assignments }.clearRupUnits;
                      (f.insert c, true)
    else (f, false)) =
    (f', true)
f'_def : f' = f.insert c
⊢ equisat (PosFin n) f f'
Tactic:
simp at ratAddSuccess
Proof state after:
n : Nat
f : DefaultFormula n
c : DefaultClause n
pivot : Literal (PosFin n)
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
f' : DefaultFormula n
f_readyForRatAdd : f.readyForRatAdd
pivot_in_c : pivot ∈ Clause.toList c
f'_def : f' = f.insert c
ratAddSuccess :
  (if f.ratHintsExhaustive pivot ratHints = true then
      if (f.insertRupUnits c.negate).snd = true then ((f.insertRupUnits c.negate).fst, false)
      else
        if ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.snd = true then
          (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, false)
        else
          if ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.fst = true then
            (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, false)
          else
            if
                (Array.foldl
                      (fun x ratHint =>
                        if x.snd = true then x.fst.performRatCheck pivot.negateLiteral ratHint else (x.fst, false))
                      (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, true) ratHints 0).snd =
                  false then
              ((Array.foldl
                    (fun x ratHint =>
                      if x.snd = true then x.fst.performRatCheck pivot.negateLiteral ratHint else (x.fst, false))
                    (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, true) ratHints 0).fst,
                false)
            else
              ({
                        clauses :=
                          (Array.foldl
                                (fun x ratHint =>
                                  if x.snd = true then x.fst.performRatCheck pivot.negateLiteral ratHint
                                  else (x.fst, false))
                                (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, true) ratHints
                                0).1.clauses,
                        rupUnits :=
                          (Array.foldl
                                (fun x ratHint =>
                                  if x.snd = true then x.fst.performRatCheck pivot.negateLiteral ratHint
                                  else (x.fst, false))
                                (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, true) ratHints
                                0).1.rupUnits,
                        ratUnits :=
                          (Array.foldl
                                (fun x ratHint =>
                                  if x.snd = true then x.fst.performRatCheck pivot.negateLiteral ratHint
                                  else (x.fst, false))
                                (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, true) ratHints
                                0).1.ratUnits,
                        assignments :=
                          restoreAssignments
                            (Array.foldl
                                  (fun x ratHint =>
                                    if x.snd = true then x.fst.performRatCheck pivot.negateLiteral ratHint
                                    else (x.fst, false))
                                  (((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, true) ratHints
                                  0).1.assignments
                            ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.fst }.clearRupUnits.insert
                  c,
                true)
    else (f, false)) =
    (f', true)
⊢ equisat (PosFin n) f f'