internlm/Lean-Github huggingface dataset:
Theorem: localRuleTruth
Proof state before:
case mpr.intro.intro.intro.intro.intro.none
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
in_ress : (X, none) ∈ ress
w_Ci :
  (M, w)⊨match Option.map Sum.inr none with
    | none => (L, R ++ X, some (Sum.inr (~'χ)))
    | some (Sum.inl (~'χ)) => (L, R ++ X, some (Sum.inl (~'χ)))
    | some (Sum.inr (~'χ)) => (L, R ++ X, some (Sum.inr (~'χ)))
⊢ evaluate M w f

case mpr.intro.intro.intro.intro.intro.some
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
val✝ : NegLoadFormula
in_ress : (X, some val✝) ∈ ress
w_Ci :
  (M, w)⊨match Option.map Sum.inr (some val✝) with
    | none => (L, R ++ X, some (Sum.inr (~'χ)))
    | some (Sum.inl (~'χ)) => (L, R ++ X, some (Sum.inl (~'χ)))
    | some (Sum.inr (~'χ)) => (L, R ++ X, some (Sum.inr (~'χ)))
⊢ evaluate M w f
Tactic:
all_goals simp at *
Proof state after:
case mpr.intro.intro.intro.intro.intro.none
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
in_ress : (X, none) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
⊢ evaluate M w f

case mpr.intro.intro.intro.intro.intro.some
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
val✝ : NegLoadFormula
in_ress : (X, some val✝) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val✝.1)))
⊢ evaluate M w f
Theorem: localRuleTruth
Proof state before:
case mpr.intro.intro.intro.intro.intro.some
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
val✝ : NegLoadFormula
in_ress : (X, some val✝) ∈ ress
w_Ci :
  (M, w)⊨match Option.map Sum.inr (some val✝) with
    | none => (L, R ++ X, some (Sum.inr (~'χ)))
    | some (Sum.inl (~'χ)) => (L, R ++ X, some (Sum.inl (~'χ)))
    | some (Sum.inr (~'χ)) => (L, R ++ X, some (Sum.inr (~'χ)))
⊢ evaluate M w f
Tactic:
simp at *
Proof state after:
case mpr.intro.intro.intro.intro.intro.some
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
val✝ : NegLoadFormula
in_ress : (X, some val✝) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val✝.1)))
⊢ evaluate M w f
Theorem: localRuleTruth
Proof state before:
case mpr.intro.intro.intro.intro.intro.none
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
in_ress : (X, none) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
⊢ evaluate M w f
Tactic:
have := w_Ci f
Proof state after:
case mpr.intro.intro.intro.intro.intro.none
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this✝ :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
in_ress : (X, none) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
this : f ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → evaluate M w f
⊢ evaluate M w f
Theorem: localRuleTruth
Proof state before:
case mpr.intro.intro.intro.intro.intro.none
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this✝ :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
in_ress : (X, none) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
this : f ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → evaluate M w f
⊢ evaluate M w f
Tactic:
aesop
Proof state after:
no goals
Theorem: localRuleTruth
Proof state before:
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
f_in : (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ evaluate M w f
Tactic:
rcases f_in with (f_in|f_in)|f_in
Proof state after:
case inl.inl
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ L
⊢ evaluate M w f

case inl.inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ R
⊢ evaluate M w f

case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : (~unload χ) = f
⊢ evaluate M w f
Theorem: localRuleTruth
Proof state before:
case inl.inl
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ L
⊢ evaluate M w f
Tactic:
apply w_Ci
Proof state after:
case inl.inl.a
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ L
⊢ f ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'val.1)))).toList
Theorem: localRuleTruth
Proof state before:
case inl.inl.a
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ L
⊢ f ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'val.1)))).toList
Tactic:
simp_all
Proof state after:
no goals
Theorem: localRuleTruth
Proof state before:
case inl.inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ R
⊢ evaluate M w f
Tactic:
apply w_Ci
Proof state after:
case inl.inr.a
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ R
⊢ f ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'val.1)))).toList
Theorem: localRuleTruth
Proof state before:
case inl.inr.a
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : f ∈ R
⊢ f ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'val.1)))).toList
Tactic:
simp_all
Proof state after:
no goals
Theorem: localRuleTruth
Proof state before:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
f : Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
f_in : (~unload χ) = f
⊢ evaluate M w f
Tactic:
subst f_in
Proof state after:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ evaluate M w (~unload χ)
Theorem: localRuleTruth
Proof state before:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ evaluate M w (~unload χ)
Tactic:
simp only [evaluate]
Proof state after:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ¬evaluate M w (unload χ)
Theorem: localRuleTruth
Proof state before:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ¬evaluate M w (unload χ)
Tactic:
rw [this]
Proof state after:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
Theorem: localRuleTruth
Proof state before:
case inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
Tactic:
use Con (X ++ Option.toList (Option.map negUnload (some val)))
Proof state after:
case h
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ (∃ a b,
      (a, b) ∈ ress ∧
        Con (a ++ (Option.map negUnload b).toList) = Con (X ++ (Option.map negUnload (some val)).toList)) ∧
    evaluate M w (Con (X ++ (Option.map negUnload (some val)).toList))
Theorem: localRuleTruth
Proof state before:
case h
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ (∃ a b,
      (a, b) ∈ ress ∧
        Con (a ++ (Option.map negUnload b).toList) = Con (X ++ (Option.map negUnload (some val)).toList)) ∧
    evaluate M w (Con (X ++ (Option.map negUnload (some val)).toList))
Tactic:
constructor
Proof state after:
case h.left
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∃ a b,
    (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = Con (X ++ (Option.map negUnload (some val)).toList)

case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ evaluate M w (Con (X ++ (Option.map negUnload (some val)).toList))
Theorem: localRuleTruth
Proof state before:
case h.left
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∃ a b,
    (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = Con (X ++ (Option.map negUnload (some val)).toList)
Tactic:
use X, some val
Proof state after:
no goals
Theorem: localRuleTruth
Proof state before:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ evaluate M w (Con (X ++ (Option.map negUnload (some val)).toList))
Tactic:
rw [conEval]
Proof state after:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∀ f ∈ X ++ (Option.map negUnload (some val)).toList, evaluate M w f
Theorem: localRuleTruth
Proof state before:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∀ f ∈ X ++ (Option.map negUnload (some val)).toList, evaluate M w f
Tactic:
simp only [Option.map_some', negUnload, Option.toList_some, List.mem_append, List.mem_singleton]
Proof state after:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∀ (f : Formula), f ∈ X ∨ f = (~unload val.1) → evaluate M w f
Theorem: localRuleTruth
Proof state before:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
⊢ ∀ (f : Formula), f ∈ X ∨ f = (~unload val.1) → evaluate M w f
Tactic:
intro g g_in
Proof state after:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
g_in : g ∈ X ∨ g = (~unload val.1)
⊢ evaluate M w g
Theorem: localRuleTruth
Proof state before:
case h.right
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
g_in : g ∈ X ∨ g = (~unload val.1)
⊢ evaluate M w g
Tactic:
rcases g_in with (_|g_def)
Proof state after:
case h.right.inl
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
h✝ : g ∈ X
⊢ evaluate M w g

case h.right.inr
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
g_def : g = (~unload val.1)
⊢ evaluate M w g
Theorem: localRuleTruth
Proof state before:
case h.right.inl
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
h✝ : g ∈ X
⊢ evaluate M w g
Tactic:
apply w_Ci
Proof state after:
case h.right.inl.a
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
h✝ : g ∈ X
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'val.1)))).toList
Theorem: localRuleTruth
Proof state before:
case h.right.inl.a
W : Type
L R : List Formula
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  ¬evaluate M w (unload χ) ↔
    ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L, R ++ X, some (Sum.inr (~'val.1)))
g : Formula
h✝ : g ∈ X
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'val.1)))).toList
Tactic:
simp_all
Proof state after:
no goals