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Proof step 9 from proof prefix-at (previous step)

st0[m] = mkblock(x, lax, m2), st1[m] = mkblock(x0, lax0, m3), @(m0, m1) ∈ lax, m ∈ st0, dom st0 ⊆ dom st1, m ∈ st1 → x ⊑ x0 ∧ lax ⊑ lax0 ∧ # x = # lax ∧ # x0 = # lax0 ⊦ x[pos(@(m0, m1), lax)] = st1[@(m, m0, m1)]

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:
st1[m] = mkblock(x0, lax0, m3), st0[m] = mkblock(x, lax, m2), m ∈ st0, @(m0, m1) ∈ lax, m ∈ st1 → x ⊑ x0 ∧ lax ⊑ lax0 ∧ # x = # lax ∧ # x0 = # lax0, ∀ n. n ∈ dom st0 → n ∈ dom st1 ⊦ x[pos(@(m0, m1), lax)] = st1[@(m, m0, m1)]

Rule argument

spec: set-basic, inst: nat-set name: Subset seq: ⊦ ns ⊆ ns0 ↔ (∀ n. n ∈ ns → n ∈ ns0) subst: [ns0, ns] → [dom st1, dom st0] flags: <all paths>

Additional information

Used terms: dom st1, dom st0