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Proof step 12 from proof nodes-write-same-flushed (previous step)

gstore[m] = groupblock(gndx, lax, n0), gstore ↓[m, groupblock(prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0)][m1] = groupblock(gndx0, lax0, m4), m ∈ gstore, valid-group-nodes(gndx) ≤ m0, n0 ≤ m0, valid-group-nodes(gndx) ≤ n0 ⊦ ( ¬ m1 ∈ ginval → ( m1 ∈ gstore ↓[m, groupblock(prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0)] ∧ @(m2, m3) ∈ lax0 ↔ @(m1, m2, m3) ∈ gstore ↓)) ∧ ( m1 ∈ gstore ↓[m, groupblock(prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0)] ∧ @(m2, m3) ∈ lax0 ∧ ¬ m1 ∈ ginval → nodes(gstore ↓[m, groupblock(prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0)], ginval)[@(m1, m2, m3)] = nodes(gstore ↓, ginval)[@(m1, m2, m3)])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ n ∈ gstore[n0, gblock] ↔ n = n0 ∨ n ∈ gstore
Proof step 14:
gstore ↓[m, groupblock(prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0)][m1] = groupblock(gndx0, lax0, m4), gstore[m] = groupblock(gndx, lax, n0), m ∈ gstore, valid-group-nodes(gndx) ≤ n0, n0 ≤ m0, valid-group-nodes(gndx) ≤ m0 ⊦ (¬ m1 ∈ ginval → ((m1 = m ∨ m1 ∈ gstore ↓) ∧ @(m2, m3) ∈ lax0 ↔ @(m1, m2, m3) ∈ gstore ↓)) ∧ ( (m1 = m ∨ m1 ∈ gstore ↓) ∧ @(m2, m3) ∈ lax0 ∧ ¬ m1 ∈ ginval → nodes(gstore ↓[m, groupblock(prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0)], ginval)[@(m1, m2, m3)] = nodes(gstore ↓, ginval)[@(m1, m2, m3)])

Rule argument

spec: store, inst: group-store-ablock-store-nat-store name: In-insert seq: ⊦ n ∈ gstore[n0, gblock] ↔ n = n0 ∨ n ∈ gstore subst: [gblock, n0, gstore, n] → [groupblock (prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0), m, gstore ↓, m1] flags: <all paths>

Additional information

Used terms: groupblock (prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0), m, gstore ↓, m1