internlm/Lean-Github huggingface dataset:
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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'val.1)))
g : Formula
h✝ : g ∈ X
⊢ g ∈ L ++ X ∪ R ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inl (~'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 ++ X, R, some (Sum.inl (~'val.1)))
g : Formula
h✝ : g ∈ X
⊢ g ∈ L ++ X ∪ R ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inl (~'val.1)))).toList
Tactic:
simp_all
Proof state after:
no goals
Theorem: localRuleTruth
Proof state before:
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 ++ X, R, some (Sum.inl (~'val.1)))
g : Formula
g_def : g = (~unload val.1)
⊢ evaluate M w g
Tactic:
subst g_def
Proof state after:
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 ++ X, R, some (Sum.inl (~'val.1)))
⊢ evaluate M w (~unload val.1)
Theorem: localRuleTruth
Proof state before:
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 ++ X, R, some (Sum.inl (~'val.1)))
⊢ evaluate M w (~unload val.1)
Tactic:
apply w_Ci
Proof state after:
case h.right.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
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L ++ X, R, some (Sum.inl (~'val.1)))
⊢ (~unload val.1) ∈ L ++ X ∪ R ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inl (~'val.1)))).toList
Theorem: localRuleTruth
Proof state before:
case h.right.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
val : NegLoadFormula
in_ress : (X, some val) ∈ ress
w_Ci : (M, w)⊨(L ++ X, R, some (Sum.inl (~'val.1)))
⊢ (~unload val.1) ∈ L ++ X ∪ R ∪ (Option.map (Sum.elim negUnload negUnload) (some (Sum.inl (~'val.1)))).toList
Tactic:
simp_all
Proof state after:
no goals
Theorem: localRuleTruth
Proof state before:
W : Type
L R : List Formula
C : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
hC : C = applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O)
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ C, (M, w)⊨Ci
Tactic:
have := loadRuleTruth lrule W M w
Proof state after:
W : Type
L R : List Formula
C : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
hC : C = applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O)
this :
  evaluate M w (~unload χ) ↔
    evaluate M w
      (dis
        (List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress))
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ C, (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
W : Type
L R : List Formula
C : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
hC : C = applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O)
this :
  evaluate M w (~unload χ) ↔
    evaluate M w
      (dis
        (List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress))
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ C, (M, w)⊨Ci
Tactic:
rw [disEval] at this
Proof state after:
W : Type
L R : List Formula
C : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
hC : C = applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O)
this :
  evaluate M w (~unload χ) ↔
    ∃
      f ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ C, (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
W : Type
L R : List Formula
C : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
hC : C = applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O)
this :
  evaluate M w (~unload χ) ↔
    ∃
      f ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ C, (M, w)⊨Ci
Tactic:
subst hC
Proof state after:
W : Type
L R : List Formula
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
this :
  evaluate M w (~unload χ) ↔
    ∃
      f ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O), (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
W : Type
L R : List Formula
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
preconditionProof : ∅.Sublist L ∧ ∅.Sublist R ∧ some (Sum.inr (~'χ)) ⊆ O
this :
  evaluate M w (~unload χ) ↔
    ∃
      f ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O), (M, w)⊨Ci
Tactic:
simp at preconditionProof
Proof state after:
W : Type
L R : List Formula
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  evaluate M w (~unload χ) ↔
    ∃
      f ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
preconditionProof : some (Sum.inr (~'χ)) = O
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O), (M, w)⊨Ci
Theorem: localRuleTruth
Proof state before:
W : Type
L R : List Formula
O : Option (NegLoadFormula ⊕ NegLoadFormula)
M : KripkeModel W
w : W
ress : List (List Formula × Option NegLoadFormula)
χ : LoadFormula
lrule : LoadRule (~'χ) ress
this :
  evaluate M w (~unload χ) ↔
    ∃
      f ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
preconditionProof : some (Sum.inr (~'χ)) = O
⊢ (M, w)⊨(L, R, O) ↔ ∃ Ci ∈ applyLocalRule (LocalRule.loadedR χ lrule) (L, R, O), (M, w)⊨Ci
Tactic:
subst preconditionProof
Proof state after:
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 ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
⊢ (M, w)⊨(L, R, some (Sum.inr (~'χ))) ↔
    ∃ Ci ∈ applyLocalRule (LocalRule.loadedR χ lrule) (L, R, some (Sum.inr (~'χ))), (M, w)⊨Ci
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 ∈
        List.map
          (fun x =>
            match x with
            | (fs, o) => Con (fs ++ (Option.map negUnload o).toList))
          ress,
      evaluate M w f
⊢ (M, w)⊨(L, R, some (Sum.inr (~'χ))) ↔
    ∃ Ci ∈ applyLocalRule (LocalRule.loadedR χ lrule) (L, R, some (Sum.inr (~'χ))), (M, w)⊨Ci
Tactic:
simp [modelCanSemImplyForm,modelCanSemImplyLLO] at *
Proof state after:
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
⊢ (∀ (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
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
⊢ (∀ (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:
constructor
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
⊢ (∀ (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

case mpr
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
⊢ (∃ 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) →
    ∀ (f : Formula), (f ∈ L ∨ f ∈ R) ∨ (~unload χ) = f → evaluate M w f
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
⊢ (∀ (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:
intro 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
⊢ ∃ 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