cases orule
α : Formula
L R precond : Finset Formula
ress✝ : List (Finset Formula)
α_in_L : α ∈ L
orule : OneSidedLocalRule precond ress✝
precond_eq : precond = {α}
c : TNode
Lcond' : Finset Formula
L_cond_in : Lcond' ⊆ L
ress : List (Finset Formula)
orule' : OneSidedLocalRule Lcond' ress
R_cond_in : ∅ ⊆ R
c_in : c ∈ applyLocalRule (LocalRule.oneSidedL orule') (L, R)
⊢ α ∈ c.1 ∨ ∃ res ∈ ress✝, res ⊆ c.1
case bot
α : Formula
L R : Finset Formula
α_in_L : α ∈ L
c : TNode
Lcond' : Finset Formula
L_cond_in : Lcond' ⊆ L
ress : List (Finset Formula)
orule' : OneSidedLocalRule Lcond' ress
R_cond_in : ∅ ⊆ R
c_in : c ∈ applyLocalRule (LocalRule.oneSidedL orule') (L, R)
precond_eq : {⊥} = {α}
⊢ α ∈ c.1 ∨ ∃ res ∈ ∅, res ⊆ c.1

case not
α : Formula
L R : Finset Formula
α_in_L : α ∈ L
c : TNode
Lcond' : Finset Formula
L_cond_in : Lcond' ⊆ L
ress : List (Finset Formula)
orule' : OneSidedLocalRule Lcond' ress
R_cond_in : ∅ ⊆ R
c_in : c ∈ applyLocalRule (LocalRule.oneSidedL orule') (L, R)
φ✝ : Formula
precond_eq : {φ✝, ~φ✝} = {α}
⊢ α ∈ c.1 ∨ ∃ res ∈ ∅, res ⊆ c.1

case neg
α : Formula
L R : Finset Formula
α_in_L : α ∈ L
c : TNode
Lcond' : Finset Formula
L_cond_in : Lcond' ⊆ L
ress : List (Finset Formula)
orule' : OneSidedLocalRule Lcond' ress
R_cond_in : ∅ ⊆ R
c_in : c ∈ applyLocalRule (LocalRule.oneSidedL orule') (L, R)
φ✝ : Formula
precond_eq : {~~φ✝} = {α}
⊢ α ∈ c.1 ∨ ∃ res ∈ [{φ✝}], res ⊆ c.1

case con
α : Formula
L R : Finset Formula
α_in_L : α ∈ L
c : TNode
Lcond' : Finset Formula
L_cond_in : Lcond' ⊆ L
ress : List (Finset Formula)
orule' : OneSidedLocalRule Lcond' ress
R_cond_in : ∅ ⊆ R
c_in : c ∈ applyLocalRule (LocalRule.oneSidedL orule') (L, R)
φ✝ ψ✝ : Formula
precond_eq : {φ✝⋀ψ✝} = {α}
⊢ α ∈ c.1 ∨ ∃ res ∈ [{φ✝, ψ✝}], res ⊆ c.1

case ncon
α : Formula
L R : Finset Formula
α_in_L : α ∈ L
c : TNode
Lcond' : Finset Formula
L_cond_in : Lcond' ⊆ L
ress : List (Finset Formula)
orule' : OneSidedLocalRule Lcond' ress
R_cond_in : ∅ ⊆ R
c_in : c ∈ applyLocalRule (LocalRule.oneSidedL orule') (L, R)
φ✝ ψ✝ : Formula
precond_eq : {~(φ✝⋀ψ✝)} = {α}
⊢ α ∈ c.1 ∨ ∃ res ∈ [{~φ✝}, {~ψ✝}], res ⊆ c.1