internlm/Lean-Github huggingface dataset:
Theorem: addRatCaseSound
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
f'_readyForRupAdd : readyForRupAdd f'
f'_readyForRatAdd : readyForRatAdd f'
ih : unsatisfiable α f'
f_equisat_f' : unsatisfiable α f ↔ unsatisfiable α f'
p : α → Bool
pf : p ⊨ f
⊢ False
Tactic:
rw [← f_equisat_f'] at ih
Proof state after:
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
f'_readyForRupAdd : readyForRupAdd f'
f'_readyForRatAdd : readyForRatAdd f'
ih : unsatisfiable α f
f_equisat_f' : unsatisfiable α f ↔ unsatisfiable α f'
p : α → Bool
pf : p ⊨ f
⊢ False
Theorem: addRatCaseSound
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
f'_readyForRupAdd : readyForRupAdd f'
f'_readyForRatAdd : readyForRatAdd f'
ih : unsatisfiable α f
f_equisat_f' : unsatisfiable α f ↔ unsatisfiable α f'
p : α → Bool
pf : p ⊨ f
⊢ False
Tactic:
exact ih p pf
Proof state after:
no goals
Theorem: addRatCaseSound
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
⊢ readyForRupAdd f'
Tactic:
rw [f'_def]
Proof state after:
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
⊢ readyForRupAdd (Formula.insert f c)
Theorem: addRatCaseSound
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
⊢ readyForRupAdd (Formula.insert f c)
Tactic:
exact insert_readyForRupAdd f c f_readyForRupAdd
Proof state after:
no goals
Theorem: addRatCaseSound
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
f'_readyForRupAdd : readyForRupAdd f'
⊢ readyForRatAdd f'
Tactic:
rw [f'_def]
Proof state after:
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
f'_readyForRupAdd : readyForRupAdd f'
⊢ readyForRatAdd (Formula.insert f c)
Theorem: addRatCaseSound
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
c : β
pivot : Literal α
f' : σ
rupHints : Array Nat
ratHints : Array (Nat × Array Nat)
pivot_limplies_c : pivot ∈ Clause.toList c
heq : performRatAdd f c pivot rupHints ratHints = (f', true)
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f'_success : lratChecker f' restPrf = success
f'_def : f' = Formula.insert f c
f'_readyForRupAdd : readyForRupAdd f'
⊢ readyForRatAdd (Formula.insert f c)
Tactic:
exact insert_readyForRatAdd f c f_readyForRatAdd
Proof state after:
no goals
Theorem: delCaseSound
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
⊢ unsatisfiable α f
Tactic:
intro p pf
Proof state after:
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
p : α → Bool
pf : p ⊨ f
⊢ False
Theorem: delCaseSound
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
p : α → Bool
pf : p ⊨ f
⊢ False
Tactic:
have f_del_readyForRupAdd : readyForRupAdd (Formula.delete f ids) := delete_readyForRupAdd f ids f_readyForRupAdd
Proof state after:
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
p : α → Bool
pf : p ⊨ f
f_del_readyForRupAdd : readyForRupAdd (Formula.delete f ids)
⊢ False
Theorem: delCaseSound
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
p : α → Bool
pf : p ⊨ f
f_del_readyForRupAdd : readyForRupAdd (Formula.delete f ids)
⊢ False
Tactic:
have f_del_readyForRatAdd : readyForRatAdd (Formula.delete f ids) := delete_readyForRatAdd f ids f_readyForRatAdd
Proof state after:
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
p : α → Bool
pf : p ⊨ f
f_del_readyForRupAdd : readyForRupAdd (Formula.delete f ids)
f_del_readyForRatAdd : readyForRatAdd (Formula.delete f ids)
⊢ False
Theorem: delCaseSound
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
ids : Array Nat
restPrf : List (Action β α)
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
h : lratChecker (Formula.delete f ids) restPrf = success
p : α → Bool
pf : p ⊨ f
f_del_readyForRupAdd : readyForRupAdd (Formula.delete f ids)
f_del_readyForRatAdd : readyForRatAdd (Formula.delete f ids)
⊢ False
Tactic:
exact ih (delete f ids) f_del_readyForRupAdd f_del_readyForRatAdd restPrfWellFormed h p (limplies_delete p pf)
Proof state after:
no goals
Theorem: lratCheckerSound
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
prf : List (Action β α)
prfWellFormed : ∀ (a : Action β α), a ∈ prf → wellFormedAction a
⊢ lratChecker f prf = success → unsatisfiable α f
Tactic:
induction prf generalizing f
Proof state after:
case nil
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
prfWellFormed : ∀ (a : Action β α), a ∈ [] → wellFormedAction a
⊢ lratChecker f [] = success → unsatisfiable α f

case cons
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
head✝ : Action β α
tail✝ : List (Action β α)
tail_ih✝ :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ tail✝ → wellFormedAction a) → lratChecker f tail✝ = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : ∀ (a : Action β α), a ∈ head✝ :: tail✝ → wellFormedAction a
⊢ lratChecker f (head✝ :: tail✝) = success → unsatisfiable α f
Theorem: lratCheckerSound
Proof state before:
case nil
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
prfWellFormed : ∀ (a : Action β α), a ∈ [] → wellFormedAction a
⊢ lratChecker f [] = success → unsatisfiable α f

case cons
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
head✝ : Action β α
tail✝ : List (Action β α)
tail_ih✝ :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ tail✝ → wellFormedAction a) → lratChecker f tail✝ = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : ∀ (a : Action β α), a ∈ head✝ :: tail✝ → wellFormedAction a
⊢ lratChecker f (head✝ :: tail✝) = success → unsatisfiable α f
Tactic:
. unfold lratChecker
  simp only [false_implies]
Proof state after:
case cons
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
head✝ : Action β α
tail✝ : List (Action β α)
tail_ih✝ :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ tail✝ → wellFormedAction a) → lratChecker f tail✝ = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : ∀ (a : Action β α), a ∈ head✝ :: tail✝ → wellFormedAction a
⊢ lratChecker f (head✝ :: tail✝) = success → unsatisfiable α f
Theorem: lratCheckerSound
Proof state before:
case cons
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
head✝ : Action β α
tail✝ : List (Action β α)
tail_ih✝ :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ tail✝ → wellFormedAction a) → lratChecker f tail✝ = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : ∀ (a : Action β α), a ∈ head✝ :: tail✝ → wellFormedAction a
⊢ lratChecker f (head✝ :: tail✝) = success → unsatisfiable α f
Tactic:
. next action restPrf ih =>
  simp only [List.find?, List.mem_cons, forall_eq_or_imp] at prfWellFormed
  rcases prfWellFormed with ⟨actionWellFormed, restPrfWellFormed⟩
  unfold lratChecker
  split
  . intro h
    exfalso
    simp only at h
  . next id rupHints restPrf' _ =>
    simp only [ite_eq_left_iff, Bool.not_eq_true]
    intro rupAddSuccess
    rw [← Bool.not_eq_true, imp_false, Classical.not_not] at rupAddSuccess
    exact addEmptyCaseSound f f_readyForRupAdd rupHints rupAddSuccess
  . next id c rupHints restPrf' hprf =>
    split
    next f' checkSuccess heq =>
    split
    . next hCheckSuccess =>
      intro f'_success
      simp only [List.cons.injEq] at hprf
      rw [← hprf.2] at f'_success
      rw [hCheckSuccess] at heq
      exact addRupCaseSound f f_readyForRupAdd f_readyForRatAdd c f' rupHints heq restPrf restPrfWellFormed ih f'_success
    . simp only [false_implies]
  . next id c pivot rupHints ratHints restPrf' hprf =>
    split
    next f' checkSuccess heq =>
    split
    . next hCheckSuccess =>
      intro f'_success
      simp only [List.cons.injEq] at hprf
      rw [← hprf.2] at f'_success
      rw [hCheckSuccess] at heq
      simp only [wellFormedAction, hprf.1] at actionWellFormed
      exact addRatCaseSound f f_readyForRupAdd f_readyForRatAdd c pivot f' rupHints ratHints actionWellFormed heq restPrf
        restPrfWellFormed ih f'_success
    . simp only [false_implies]
  . next ids restPrf' hprf =>
    intro h
    simp only [List.cons.injEq] at hprf
    rw [← hprf.2] at h
    exact delCaseSound f f_readyForRupAdd f_readyForRatAdd ids restPrf restPrfWellFormed ih h
Proof state after:
no goals
Theorem: lratCheckerSound
Proof state before:
case nil
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
prfWellFormed : ∀ (a : Action β α), a ∈ [] → wellFormedAction a
⊢ lratChecker f [] = success → unsatisfiable α f
Tactic:
unfold lratChecker
Proof state after:
case nil
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
prfWellFormed : ∀ (a : Action β α), a ∈ [] → wellFormedAction a
⊢ out_of_proof = success → unsatisfiable α f
Theorem: lratCheckerSound
Proof state before:
case nil
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
prfWellFormed : ∀ (a : Action β α), a ∈ [] → wellFormedAction a
⊢ out_of_proof = success → unsatisfiable α f
Tactic:
simp only [false_implies]
Proof state after:
no goals
Theorem: lratCheckerSound
Proof state before:
case cons
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
head✝ : Action β α
tail✝ : List (Action β α)
tail_ih✝ :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ tail✝ → wellFormedAction a) → lratChecker f tail✝ = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : ∀ (a : Action β α), a ∈ head✝ :: tail✝ → wellFormedAction a
⊢ lratChecker f (head✝ :: tail✝) = success → unsatisfiable α f
Tactic:
next action restPrf ih =>
simp only [List.find?, List.mem_cons, forall_eq_or_imp] at prfWellFormed
rcases prfWellFormed with ⟨actionWellFormed, restPrfWellFormed⟩
unfold lratChecker
split
. intro h
  exfalso
  simp only at h
. next id rupHints restPrf' _ =>
  simp only [ite_eq_left_iff, Bool.not_eq_true]
  intro rupAddSuccess
  rw [← Bool.not_eq_true, imp_false, Classical.not_not] at rupAddSuccess
  exact addEmptyCaseSound f f_readyForRupAdd rupHints rupAddSuccess
. next id c rupHints restPrf' hprf =>
  split
  next f' checkSuccess heq =>
  split
  . next hCheckSuccess =>
    intro f'_success
    simp only [List.cons.injEq] at hprf
    rw [← hprf.2] at f'_success
    rw [hCheckSuccess] at heq
    exact addRupCaseSound f f_readyForRupAdd f_readyForRatAdd c f' rupHints heq restPrf restPrfWellFormed ih f'_success
  . simp only [false_implies]
. next id c pivot rupHints ratHints restPrf' hprf =>
  split
  next f' checkSuccess heq =>
  split
  . next hCheckSuccess =>
    intro f'_success
    simp only [List.cons.injEq] at hprf
    rw [← hprf.2] at f'_success
    rw [hCheckSuccess] at heq
    simp only [wellFormedAction, hprf.1] at actionWellFormed
    exact addRatCaseSound f f_readyForRupAdd f_readyForRatAdd c pivot f' rupHints ratHints actionWellFormed heq restPrf
      restPrfWellFormed ih f'_success
  . simp only [false_implies]
. next ids restPrf' hprf =>
  intro h
  simp only [List.cons.injEq] at hprf
  rw [← hprf.2] at h
  exact delCaseSound f f_readyForRupAdd f_readyForRatAdd ids restPrf restPrfWellFormed ih h
Proof state after:
no goals
Theorem: lratCheckerSound
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
action : Action β α
restPrf : List (Action β α)
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : ∀ (a : Action β α), a ∈ action :: restPrf → wellFormedAction a
⊢ lratChecker f (action :: restPrf) = success → unsatisfiable α f
Tactic:
simp only [List.find?, List.mem_cons, forall_eq_or_imp] at prfWellFormed
Proof state after:
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
action : Action β α
restPrf : List (Action β α)
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : wellFormedAction action ∧ ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
⊢ lratChecker f (action :: restPrf) = success → unsatisfiable α f
Theorem: lratCheckerSound
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
action : Action β α
restPrf : List (Action β α)
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
prfWellFormed : wellFormedAction action ∧ ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
⊢ lratChecker f (action :: restPrf) = success → unsatisfiable α f
Tactic:
rcases prfWellFormed with ⟨actionWellFormed, restPrfWellFormed⟩
Proof state after:
case intro
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
action : Action β α
restPrf : List (Action β α)
ih :
  ∀ (f : σ),
    readyForRupAdd f →
      readyForRatAdd f →
        (∀ (a : Action β α), a ∈ restPrf → wellFormedAction a) → lratChecker f restPrf = success → unsatisfiable α f
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
actionWellFormed : wellFormedAction action
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → wellFormedAction a
⊢ lratChecker f (action :: restPrf) = success → unsatisfiable α f