internlm/Lean-Github huggingface dataset:
Theorem: localRuleTruth
Proof state before:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
have hyp' := hyp (~unload χ)
Proof state after:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
hyp' : ((~unload χ) ∈ L ∨ (~unload χ) ∈ R) ∨ (~unload χ) = (~unload χ) → evaluate M w (~unload χ)
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
hyp' : ((~unload χ) ∈ L ∨ (~unload χ) ∈ R) ∨ (~unload χ) = (~unload χ) → evaluate M w (~unload χ)
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
simp at hyp'
Proof state after:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
hyp' : ¬evaluate M w (unload χ)
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
hyp' : ¬evaluate M w (unload χ)
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
rw [this] at hyp'
Proof state after:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
hyp' : ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
case mp
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
hyp' : ∃ f, (∃ a b, (a, b) ∈ ress ∧ Con (a ++ (Option.map negUnload b).toList) = f) ∧ evaluate M w f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
rcases hyp' with ⟨f, ⟨X , O, in_ress, def_f⟩, w_f⟩
Proof state after:
case mp.intro.intro.intro.intro.intro
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
O : Option NegLoadFormula
in_ress : (X, O) ∈ ress
def_f : Con (X ++ (Option.map negUnload O).toList) = f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
case mp.intro.intro.intro.intro.intro
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
O : Option NegLoadFormula
in_ress : (X, O) ∈ ress
def_f : Con (X ++ (Option.map negUnload O).toList) = f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
cases O
Proof state after:
case mp.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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci

case mp.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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
val✝ : NegLoadFormula
in_ress : (X, some val✝) ∈ ress
def_f : Con (X ++ (Option.map negUnload (some val✝)).toList) = f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
case mp.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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
use (L, R ++ X, some (Sum.inr (~'χ)))
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (∃ a b,
      (a, b) ∈ ress ∧
        (match Option.map Sum.inr b with
          | none => (L, R ++ a, some (Sum.inr (~'χ)))
          | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
          | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
          (L, R ++ X, some (Sum.inr (~'χ)))) ∧
    (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (∃ a b,
      (a, b) ∈ ress ∧
        (match Option.map Sum.inr b with
          | none => (L, R ++ a, some (Sum.inr (~'χ)))
          | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
          | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
          (L, R ++ X, some (Sum.inr (~'χ)))) ∧
    (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ ∃ a b,
    (a, b) ∈ ress ∧
      (match Option.map Sum.inr b with
        | none => (L, R ++ a, some (Sum.inr (~'χ)))
        | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
        | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
        (L, R ++ X, some (Sum.inr (~'χ)))

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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ ∃ a b,
    (a, b) ∈ ress ∧
      (match Option.map Sum.inr b with
        | none => (L, R ++ a, some (Sum.inr (~'χ)))
        | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
        | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
        (L, R ++ X, some (Sum.inr (~'χ)))
Tactic:
use X, none
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (X, none) ∈ ress ∧
    (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)))) =
      (L, R ++ X, some (Sum.inr (~'χ)))
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (X, none) ∈ ress ∧
    (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)))) =
      (L, R ++ X, some (Sum.inr (~'χ)))
Tactic:
simp only [Option.map_none', and_true]
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (X, none) ∈ ress
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
⊢ (M, w)⊨(L, R ++ X, some (Sum.inr (~'χ)))
Tactic:
intro g
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
g : Formula
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → 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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
def_f : Con (X ++ (Option.map negUnload none).toList) = f
g : Formula
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → evaluate M w g
Tactic:
subst def_f
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
g : Formula
w_f : evaluate M w (Con (X ++ (Option.map negUnload none).toList))
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → 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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
g : Formula
w_f : evaluate M w (Con (X ++ (Option.map negUnload none).toList))
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → evaluate M w g
Tactic:
rw [conEval] at w_f
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
g : Formula
w_f : ∀ f ∈ X ++ (Option.map negUnload none).toList, evaluate M w f
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → 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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
X : List Formula
in_ress : (X, none) ∈ ress
g : Formula
w_f : ∀ f ∈ X ++ (Option.map negUnload none).toList, evaluate M w f
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → evaluate M w g
Tactic:
specialize hyp g
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
in_ress : (X, none) ∈ ress
g : Formula
w_f : ∀ f ∈ X ++ (Option.map negUnload none).toList, evaluate M w f
hyp : (g ∈ L ∨ g ∈ R) ∨ (~unload χ) = g → evaluate M w g
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → 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
in_ress : (X, none) ∈ ress
g : Formula
w_f : ∀ f ∈ X ++ (Option.map negUnload none).toList, evaluate M w f
hyp : (g ∈ L ∨ g ∈ R) ∨ (~unload χ) = g → evaluate M w g
⊢ g ∈ L ∪ (R ++ X) ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inr (~'χ)))).toList → evaluate M w g
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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
def_f : Con (X ++ (Option.map negUnload (some val)).toList) = f
⊢ ∃ Ci,
    (∃ a b,
        (a, b) ∈ ress ∧
          (match Option.map Sum.inr b with
            | none => (L, R ++ a, some (Sum.inr (~'χ)))
            | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
            | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
            Ci) ∧
      (M, w)⊨Ci
Tactic:
use (L, R ++ X, some (Sum.inr 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
hyp : ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
f : Formula
w_f : evaluate M w f
X : List Formula
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
def_f : Con (X ++ (Option.map negUnload (some val)).toList) = f
⊢ (∃ a b,
      (a, b) ∈ ress ∧
        (match Option.map Sum.inr b with
          | none => (L, R ++ a, some (Sum.inr (~'χ)))
          | some (Sum.inl (~'χ)) => (L, R ++ a, some (Sum.inl (~'χ)))
          | some (Sum.inr (~'χ)) => (L, R ++ a, some (Sum.inr (~'χ)))) =
          (L, R ++ X, some (Sum.inr val))) ∧
    (M, w)⊨(L, R ++ X, some (Sum.inr val))