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Proof step 9 from proof gstore-dom-eq-hole (previous step)

gstore0 \ (∅ ++ n) = gstore1 \ (∅ ++ n), dom gstore0 = dom gstore1, n ∈ gstore0 ⊦ (n0 ∈ gstore1 ↔ n0 = n ∨ n0 ∈ gstore0) ∧ (n0 ∈ gstore1 → gstore1[n0] = gstore0[n, gstore1[n]][n0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ns = ns0 ↔ (∀ n. n ∈ ns ↔ n ∈ ns0)
Proof step 11:
gstore0 \ (∅ ++ n) = gstore1 \ (∅ ++ n), n ∈ gstore0, ∀ n. n ∈ dom gstore0 ↔ n ∈ dom gstore1 ⊦ (n0 ∈ gstore1 ↔ n0 = n ∨ n0 ∈ gstore0) ∧ (n0 ∈ gstore1 → gstore1[n0] = gstore0[n, gstore1[n]][n0])

Rule argument

spec: set-basic, inst: nat-set name: Extension seq: ⊦ ns = ns0 ↔ (∀ n. n ∈ ns ↔ n ∈ ns0) subst: [ns0, ns] → [dom gstore1, dom gstore0] flags: <all paths>

Additional information

Used terms: dom gstore1, dom gstore0