unfold MultisetLT'
case mk.intro.mk.mk.intro
L R : List Formula
ress : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
Lcond Rcond : List Formula
Ocond : Option (NegLoadFormula ⊕ NegLoadFormula)
rule : LocalRule (Lcond, Rcond, Ocond) ress
preconP : Lcond.Sublist L ∧ Rcond.Sublist R ∧ Ocond ⊆ O
RES : TNode
Lnew Rnew : List Formula
Onew : Option (NegLoadFormula ⊕ NegLoadFormula)
Y_in_ress : (Lnew, Rnew, Onew) ∈ ress
claim :
  (match Onew with
    | none => (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, O)
    | some (Sum.inl (~'χ)) => (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inl (~'χ)))
    | some (Sum.inr (~'χ)) => (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inr (~'χ)))) =
    RES
⊢ MultisetLT' (node_to_multiset RES) (node_to_multiset (L, R, O))
case mk.intro.mk.mk.intro
L R : List Formula
ress : List TNode
O : Option (NegLoadFormula ⊕ NegLoadFormula)
Lcond Rcond : List Formula
Ocond : Option (NegLoadFormula ⊕ NegLoadFormula)
rule : LocalRule (Lcond, Rcond, Ocond) ress
preconP : Lcond.Sublist L ∧ Rcond.Sublist R ∧ Ocond ⊆ O
RES : TNode
Lnew Rnew : List Formula
Onew : Option (NegLoadFormula ⊕ NegLoadFormula)
Y_in_ress : (Lnew, Rnew, Onew) ∈ ress
claim :
  (match Onew with
    | none => (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, O)
    | some (Sum.inl (~'χ)) => (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inl (~'χ)))
    | some (Sum.inr (~'χ)) => (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inr (~'χ)))) =
    RES
⊢ ∃ X Y Z, Y ≠ ∅ ∧ node_to_multiset RES = Z + X ∧ node_to_multiset (L, R, O) = Z + Y ∧ ∀ x ∈ X, ∃ y ∈ Y, x < y