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Proof step 18 from proof ADD-lemma (previous step)

w ∈ S, w ∈ s, s ⊆ S ⊦ (N0 ∪ (S \ (s ∪ V0))) ++ w = N0 ∪ (S \ ((s -- w) ∪ V0)), # S = 0, # s = 0, s = ∅, S = ∅, w ∈ V0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)
Proof step 17:
w ∈ S, w ∈ s, s ⊆ S ⊦ ∀ a. a ∈ (N0 ∪ (S \ (s ∪ V0))) ++ w ↔ a ∈ N0 ∪ (S \ ((s -- w) ∪ V0)), # S = 0, # s = 0, s = ∅, S = ∅, w ∈ V0

Rule argument

[x0lemmaarg: spec-name: set-union instance-name: lemma-name: ext antecedent: succedent: s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2) substlist: [s2, s1] → [N0 ∪ (S \ ((s -- w) ∪ V0)), (N0 ∪ (S \ (s ∪ V0))) ++ w] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (N0 ∪ (S \ ((s -- w) ∪ V0)), (N0 ∪ (S \ (s ∪ V0))) ++ w)]]