internlm/Lean-Github huggingface dataset:
Theorem: incrementalLRATCheckerSound
Proof state before:
case right
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
⊢ (incrementalLRATChecker f a).snd = out_of_proof → unsatisfiable α (incrementalLRATChecker f a).fst → unsatisfiable α f
Tactic:
. intro incrementalChecker_res incrementalChecker_res_unsat
  unfold incrementalLRATChecker at incrementalChecker_res
  simp only at incrementalChecker_res
  split at incrementalChecker_res
  . next rupHints =>
    by_cases h : (performRupAdd f empty rupHints).2
    . simp only [h, ite_true] at incrementalChecker_res
    . simp only [Bool.not_eq_true] at h
      simp only [h, ite_false] at incrementalChecker_res
  . next c rupHints =>
    by_cases performRupAdd_success : (performRupAdd f c rupHints).2
    . let f' := (performRupAdd f c rupHints).1
      have heq : performRupAdd f c rupHints = (f', true) := by rw [← performRupAdd_success]
      have f_liff_f' := rupAdd_sound f c rupHints f' f_readyForRupAdd heq
      unfold incrementalLRATChecker at incrementalChecker_res_unsat
      simp only [performRupAdd_success, ite_true] at incrementalChecker_res_unsat
      rw [liff_unsat f f' f_liff_f']
      exact incrementalChecker_res_unsat
    . simp only [Bool.not_eq_true] at performRupAdd_success
      simp only [performRupAdd_success, ite_false] at incrementalChecker_res
  . next c pivot rupHints ratHints =>
    by_cases performRatAdd_success : (performRatAdd f c pivot rupHints ratHints).2
    . let f' := (performRatAdd f c pivot rupHints ratHints).1
      have heq : performRatAdd f c pivot rupHints ratHints = (f', true) := by rw [← performRatAdd_success]
      simp only [wellFormedAction, limplies_iff_mem] at aWellFormed
      unfold incrementalLRATChecker at incrementalChecker_res_unsat
      simp only [performRatAdd_success, ite_true] at incrementalChecker_res_unsat
      have h := ratAdd_sound f c pivot rupHints ratHints f' f_readyForRatAdd aWellFormed heq
      rw [equisat] at h
      rw [h]
      exact incrementalChecker_res_unsat
    . simp only [Bool.not_eq_true] at performRatAdd_success
      simp only [performRatAdd_success, ite_false] at incrementalChecker_res
  . next ids =>
    unfold incrementalLRATChecker at incrementalChecker_res_unsat
    simp only at incrementalChecker_res_unsat
    intro p pf
    exact incrementalChecker_res_unsat p (limplies_delete p pf)
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
case left
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
⊢ (incrementalLRATChecker f a).snd = success → unsatisfiable α f
Tactic:
intro incrementalChecker_success p pf
Proof state after:
case left
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
incrementalChecker_success : (incrementalLRATChecker f a).snd = success
p : α → Bool
pf : p ⊨ f
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case left
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
incrementalChecker_success : (incrementalLRATChecker f a).snd = success
p : α → Bool
pf : p ⊨ f
⊢ False
Tactic:
unfold incrementalLRATChecker at incrementalChecker_success
Proof state after:
case left
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
incrementalChecker_success :
  (match a with
      | Action.addEmpty id rupHints =>
        match performRupAdd f empty rupHints with
        | (f, checkSuccess) => if checkSuccess = true then (f, success) else (f, rup_failure)
      | Action.addRup id c rupHints =>
        match performRupAdd f c rupHints with
        | (f, checkSuccess) => if checkSuccess = true then (f, out_of_proof) else (f, rup_failure)
      | Action.addRat id c pivot rupHints ratHints =>
        match performRatAdd f c pivot rupHints ratHints with
        | (f, checkSuccess) => if checkSuccess = true then (f, out_of_proof) else (f, rup_failure)
      | Action.del ids => (Formula.delete f ids, out_of_proof)).snd =
    success
p : α → Bool
pf : p ⊨ f
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case left
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
incrementalChecker_success :
  (match a with
      | Action.addEmpty id rupHints =>
        match performRupAdd f empty rupHints with
        | (f, checkSuccess) => if checkSuccess = true then (f, success) else (f, rup_failure)
      | Action.addRup id c rupHints =>
        match performRupAdd f c rupHints with
        | (f, checkSuccess) => if checkSuccess = true then (f, out_of_proof) else (f, rup_failure)
      | Action.addRat id c pivot rupHints ratHints =>
        match performRatAdd f c pivot rupHints ratHints with
        | (f, checkSuccess) => if checkSuccess = true then (f, out_of_proof) else (f, rup_failure)
      | Action.del ids => (Formula.delete f ids, out_of_proof)).snd =
    success
p : α → Bool
pf : p ⊨ f
⊢ False
Tactic:
simp only at incrementalChecker_success
Proof state after:
case left
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
a : Action β α
aWellFormed : wellFormedAction a
incrementalChecker_success :
  (match a with
      | Action.addEmpty id rupHints =>
        if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
        else ((performRupAdd f empty rupHints).fst, rup_failure)
      | Action.addRup id c rupHints =>
        if (performRupAdd f c rupHints).snd = true then ((performRupAdd f c rupHints).fst, out_of_proof)
        else ((performRupAdd f c rupHints).fst, rup_failure)
      | Action.addRat id c pivot rupHints ratHints =>
        if (performRatAdd f c pivot rupHints ratHints).snd = true then
          ((performRatAdd f c pivot rupHints ratHints).fst, out_of_proof)
        else ((performRatAdd f c pivot rupHints ratHints).fst, rup_failure)
      | Action.del ids => (Formula.delete f ids, out_of_proof)).snd =
    success
p : α → Bool
pf : p ⊨ f
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case left.h_2
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c✝ : β
rupHints✝ : Array Nat
aWellFormed : wellFormedAction (Action.addRup id✝ c✝ rupHints✝)
incrementalChecker_success :
  (if (performRupAdd f c✝ rupHints✝).snd = true then ((performRupAdd f c✝ rupHints✝).fst, out_of_proof)
      else ((performRupAdd f c✝ rupHints✝).fst, rup_failure)).snd =
    success
⊢ False

case left.h_3
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c✝ : β
pivot✝ : Literal α
rupHints✝ : Array Nat
ratHints✝ : Array (Nat × Array Nat)
aWellFormed : wellFormedAction (Action.addRat id✝ c✝ pivot✝ rupHints✝ ratHints✝)
incrementalChecker_success :
  (if (performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).snd = true then
        ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, out_of_proof)
      else ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, rup_failure)).snd =
    success
⊢ False

case left.h_4
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_success : (Formula.delete f ids✝, out_of_proof).snd = success
⊢ False
Tactic:
. next c rupHints =>
  by_cases performRupAdd_success : (performRupAdd f c rupHints).2
  . simp only [performRupAdd_success, ite_true] at incrementalChecker_success
  . simp only [performRupAdd_success, ite_false] at incrementalChecker_success
Proof state after:
case left.h_3
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c✝ : β
pivot✝ : Literal α
rupHints✝ : Array Nat
ratHints✝ : Array (Nat × Array Nat)
aWellFormed : wellFormedAction (Action.addRat id✝ c✝ pivot✝ rupHints✝ ratHints✝)
incrementalChecker_success :
  (if (performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).snd = true then
        ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, out_of_proof)
      else ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, rup_failure)).snd =
    success
⊢ False

case left.h_4
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_success : (Formula.delete f ids✝, out_of_proof).snd = success
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case left.h_3
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c✝ : β
pivot✝ : Literal α
rupHints✝ : Array Nat
ratHints✝ : Array (Nat × Array Nat)
aWellFormed : wellFormedAction (Action.addRat id✝ c✝ pivot✝ rupHints✝ ratHints✝)
incrementalChecker_success :
  (if (performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).snd = true then
        ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, out_of_proof)
      else ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, rup_failure)).snd =
    success
⊢ False

case left.h_4
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_success : (Formula.delete f ids✝, out_of_proof).snd = success
⊢ False
Tactic:
. next c pivot rupHints ratHints =>
  by_cases performRatAdd_success : (performRatAdd f c pivot rupHints ratHints).2
  . simp only [performRatAdd_success, ite_true] at incrementalChecker_success
  . simp only [performRatAdd_success, ite_false] at incrementalChecker_success
Proof state after:
case left.h_4
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_success : (Formula.delete f ids✝, out_of_proof).snd = success
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case left.h_4
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_success : (Formula.delete f ids✝, out_of_proof).snd = success
⊢ False
Tactic:
. simp only at incrementalChecker_success
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
case left.h_1
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints✝ : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints✝)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints✝).snd = true then ((performRupAdd f empty rupHints✝).fst, success)
      else ((performRupAdd f empty rupHints✝).fst, rup_failure)).snd =
    success
⊢ False
Tactic:
next rupHints =>
by_cases performRupAdd_success : (performRupAdd f empty rupHints).2
. exact addEmptyCaseSound f f_readyForRupAdd rupHints performRupAdd_success p pf
. simp only [performRupAdd_success, ite_false] at incrementalChecker_success
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
⊢ False
Tactic:
by_cases performRupAdd_success : (performRupAdd f empty rupHints).2
Proof state after:
case pos
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : (performRupAdd f empty rupHints).snd = true
⊢ False

case neg
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : ¬(performRupAdd f empty rupHints).snd = true
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case pos
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : (performRupAdd f empty rupHints).snd = true
⊢ False

case neg
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : ¬(performRupAdd f empty rupHints).snd = true
⊢ False
Tactic:
. exact addEmptyCaseSound f f_readyForRupAdd rupHints performRupAdd_success p pf
Proof state after:
case neg
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : ¬(performRupAdd f empty rupHints).snd = true
⊢ False
Theorem: incrementalLRATCheckerSound
Proof state before:
case neg
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : ¬(performRupAdd f empty rupHints).snd = true
⊢ False
Tactic:
. simp only [performRupAdd_success, ite_false] at incrementalChecker_success
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
case pos
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : (performRupAdd f empty rupHints).snd = true
⊢ False
Tactic:
exact addEmptyCaseSound f f_readyForRupAdd rupHints performRupAdd_success p pf
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
case neg
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addEmpty id✝ rupHints)
incrementalChecker_success :
  (if (performRupAdd f empty rupHints).snd = true then ((performRupAdd f empty rupHints).fst, success)
      else ((performRupAdd f empty rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : ¬(performRupAdd f empty rupHints).snd = true
⊢ False
Tactic:
simp only [performRupAdd_success, ite_false] at incrementalChecker_success
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
case left.h_2
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c✝ : β
rupHints✝ : Array Nat
aWellFormed : wellFormedAction (Action.addRup id✝ c✝ rupHints✝)
incrementalChecker_success :
  (if (performRupAdd f c✝ rupHints✝).snd = true then ((performRupAdd f c✝ rupHints✝).fst, out_of_proof)
      else ((performRupAdd f c✝ rupHints✝).fst, rup_failure)).snd =
    success
⊢ False
Tactic:
next c rupHints =>
by_cases performRupAdd_success : (performRupAdd f c rupHints).2
. simp only [performRupAdd_success, ite_true] at incrementalChecker_success
. simp only [performRupAdd_success, ite_false] at incrementalChecker_success
Proof state after:
no goals
Theorem: incrementalLRATCheckerSound
Proof state before:
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c : β
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addRup id✝ c rupHints)
incrementalChecker_success :
  (if (performRupAdd f c rupHints).snd = true then ((performRupAdd f c rupHints).fst, out_of_proof)
      else ((performRupAdd f c rupHints).fst, rup_failure)).snd =
    success
⊢ False
Tactic:
by_cases performRupAdd_success : (performRupAdd f c rupHints).2
Proof state after:
case pos
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c : β
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addRup id✝ c rupHints)
incrementalChecker_success :
  (if (performRupAdd f c rupHints).snd = true then ((performRupAdd f c rupHints).fst, out_of_proof)
      else ((performRupAdd f c rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : (performRupAdd f c rupHints).snd = true
⊢ False

case neg
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
p : α → Bool
pf : p ⊨ f
action✝ : Action β α
id✝ : Nat
c : β
rupHints : Array Nat
aWellFormed : wellFormedAction (Action.addRup id✝ c rupHints)
incrementalChecker_success :
  (if (performRupAdd f c rupHints).snd = true then ((performRupAdd f c rupHints).fst, out_of_proof)
      else ((performRupAdd f c rupHints).fst, rup_failure)).snd =
    success
performRupAdd_success : ¬(performRupAdd f c rupHints).snd = true
⊢ False