internlm/Lean-Github huggingface dataset:
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_1
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew)
⊢ node_to_multiset (Lcond, Rcond, Ocond) ≠ ∅
Tactic:
exact (LocalRule.cond_non_empty rule : node_to_multiset (Lcond, Rcond, Ocond) ≠ ∅)
Proof state after:
no goals
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew)
⊢ node_to_multiset RES = Z + node_to_multiset (Lnew, Rnew, Onew)
Tactic:
rw [Z_eq]
Proof state after:
case h.refine_2.refine_1
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew)
⊢ node_to_multiset RES =
    node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew) + node_to_multiset (Lnew, Rnew, Onew)
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew)
⊢ node_to_multiset RES =
    node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew) + node_to_multiset (Lnew, Rnew, Onew)
Tactic:
apply Multiset.sub_add_of_subset_eq
Proof state after:
case h.refine_2.refine_1.h
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew)
⊢ node_to_multiset (Lnew, Rnew, Onew) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, Onew)
⊢ node_to_multiset (Lnew, Rnew, Onew) ≤ node_to_multiset RES
Tactic:
cases Onew
Proof state after:
case h.refine_2.refine_1.h.none
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
Y_in_ress : (Lnew, Rnew, none) ∈ ress
claim :
  (match none 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, none)
⊢ node_to_multiset (Lnew, Rnew, none) ≤ node_to_multiset RES

case h.refine_2.refine_1.h.some
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula ⊕ NegLoadFormula
Y_in_ress : (Lnew, Rnew, some val✝) ∈ ress
claim :
  (match some val✝ 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some val✝)
⊢ node_to_multiset (Lnew, Rnew, some val✝) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h.some
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula ⊕ NegLoadFormula
Y_in_ress : (Lnew, Rnew, some val✝) ∈ ress
claim :
  (match some val✝ 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some val✝)
⊢ node_to_multiset (Lnew, Rnew, some val✝) ≤ node_to_multiset RES
Tactic:
try (rename_i f; cases f)
Proof state after:
case h.refine_2.refine_1.h.some.inl
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inl val✝)) ∈ ress
claim :
  (match some (Sum.inl val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inl val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inl val✝)) ≤ node_to_multiset RES

case h.refine_2.refine_1.h.some.inr
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim :
  (match some (Sum.inr val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h.some
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula ⊕ NegLoadFormula
Y_in_ress : (Lnew, Rnew, some val✝) ∈ ress
claim :
  (match some val✝ 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some val✝)
⊢ node_to_multiset (Lnew, Rnew, some val✝) ≤ node_to_multiset RES
Tactic:
(rename_i f; cases f)
Proof state after:
case h.refine_2.refine_1.h.some.inl
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inl val✝)) ∈ ress
claim :
  (match some (Sum.inl val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inl val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inl val✝)) ≤ node_to_multiset RES

case h.refine_2.refine_1.h.some.inr
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim :
  (match some (Sum.inr val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h.some
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula ⊕ NegLoadFormula
Y_in_ress : (Lnew, Rnew, some val✝) ∈ ress
claim :
  (match some val✝ 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some val✝)
⊢ node_to_multiset (Lnew, Rnew, some val✝) ≤ node_to_multiset RES
Tactic:
rename_i f
Proof state after:
case h.refine_2.refine_1.h.some
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
f : NegLoadFormula ⊕ NegLoadFormula
Y_in_ress : (Lnew, Rnew, some f) ∈ ress
claim :
  (match some f 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some f)
⊢ node_to_multiset (Lnew, Rnew, some f) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h.some
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
f : NegLoadFormula ⊕ NegLoadFormula
Y_in_ress : (Lnew, Rnew, some f) ∈ ress
claim :
  (match some f 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some f)
⊢ node_to_multiset (Lnew, Rnew, some f) ≤ node_to_multiset RES
Tactic:
cases f
Proof state after:
case h.refine_2.refine_1.h.some.inl
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inl val✝)) ∈ ress
claim :
  (match some (Sum.inl val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inl val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inl val✝)) ≤ node_to_multiset RES

case h.refine_2.refine_1.h.some.inr
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim :
  (match some (Sum.inr val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h.some.inr
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
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim :
  (match some (Sum.inr val✝) 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
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES
Tactic:
simp_all
Proof state after:
case h.refine_2.refine_1.h.some.inr
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
RES : TNode
Lnew Rnew : List Formula
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
preconP :
  Lcond.Sublist L ∧
    Rcond.Sublist R ∧
      match Ocond, O with
      | none, x => True
      | some val, none => False
      | some f, some g => f = g
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim : (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inr (~'val✝.1))) = RES
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES
Theorem: localRuleApp.decreases_DM
Proof state before:
case h.refine_2.refine_1.h.some.inr
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
RES : TNode
Lnew Rnew : List Formula
Z : Multiset Formula := node_to_multiset (L, R, O) - node_to_multiset (Lcond, Rcond, Ocond)
val✝ : NegLoadFormula
preconP :
  Lcond.Sublist L ∧
    Rcond.Sublist R ∧
      match Ocond, O with
      | none, x => True
      | some val, none => False
      | some f, some g => f = g
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim : (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inr (~'val✝.1))) = RES
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES
Tactic:
cases O
Proof state after:
case h.refine_2.refine_1.h.some.inr.none
L R : List Formula
ress : List TNode
Lcond Rcond : List Formula
Ocond : Option (NegLoadFormula ⊕ NegLoadFormula)
rule : LocalRule (Lcond, Rcond, Ocond) ress
RES : TNode
Lnew Rnew : List Formula
val✝ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝)) ∈ ress
claim : (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inr (~'val✝.1))) = RES
Z : Multiset Formula := node_to_multiset (L, R, none) - node_to_multiset (Lcond, Rcond, Ocond)
preconP :
  Lcond.Sublist L ∧
    Rcond.Sublist R ∧
      match Ocond, none with
      | none, x => True
      | some val, none => False
      | some f, some g => f = g
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝)) ≤ node_to_multiset RES

case h.refine_2.refine_1.h.some.inr.some
L R : List Formula
ress : List TNode
Lcond Rcond : List Formula
Ocond : Option (NegLoadFormula ⊕ NegLoadFormula)
rule : LocalRule (Lcond, Rcond, Ocond) ress
RES : TNode
Lnew Rnew : List Formula
val✝¹ : NegLoadFormula
Y_in_ress : (Lnew, Rnew, some (Sum.inr val✝¹)) ∈ ress
claim : (L.diff Lcond ++ Lnew, R.diff Rcond ++ Rnew, some (Sum.inr (~'val✝¹.1))) = RES
val✝ : NegLoadFormula ⊕ NegLoadFormula
Z : Multiset Formula := node_to_multiset (L, R, some val✝) - node_to_multiset (Lcond, Rcond, Ocond)
preconP :
  Lcond.Sublist L ∧
    Rcond.Sublist R ∧
      match Ocond, some val✝ with
      | none, x => True
      | some val, none => False
      | some f, some g => f = g
Z_eq : Z = node_to_multiset RES - node_to_multiset (Lnew, Rnew, some (Sum.inr val✝¹))
⊢ node_to_multiset (Lnew, Rnew, some (Sum.inr val✝¹)) ≤ node_to_multiset RES