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Proof step 22 from proof abs-log-gcrash-down (previous step)

gstore[lnum] = groupblock(gndx0, lax0, m0), crash(gstore, gstore0, ginval, nx0) ↓[lnum] = groupblock(gndx, lax, m), ginval ∩ elems nx1 = ∅, inv(gstore \ ginval), inv(groupblock(gndx0, lax0, m0)), lnum ∈ gstore, ∀ ginval, gstore, gstore0, nx0. elems nx1 ∩ ginval = ∅ ∧ inv(gstore \ ginval) → abs-log(nx1, crash(gstore, gstore0, ginval, nx0) ↓) = abs-log(nx1, gstore ↓) ⊦ lax.to-adr(lnum) + abs-log(nx1, crash(gstore, gstore0, ginval, nx0) ↓) = prefixn(valid-group-nodes(gndx0), lax0).to-adr(lnum) + abs-log(nx1, gstore ↓), lnum ∈ ginval

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ n ∈ gstore → gstore ↓[n] = gstore[n] ↓
Proof step 24:
gstore[lnum] = groupblock(gndx0, lax0, m0), crash(gstore, gstore0, ginval, nx0)[lnum] ↓ = groupblock(gndx, lax, m), ginval ∩ elems nx1 = ∅, inv(groupblock(gndx0, lax0, m0)), inv(gstore \ ginval), lnum ∈ gstore, lnum ∈ crash(gstore, gstore0, ginval, nx0), ∀ ginval, gstore, gstore0, nx0. elems nx1 ∩ ginval = ∅ ∧ inv(gstore \ ginval) → abs-log(nx1, crash(gstore, gstore0, ginval, nx0) ↓) = abs-log(nx1, gstore ↓) ⊦ lax.to-adr(lnum) + abs-log(nx1, crash(gstore, gstore0, ginval, nx0) ↓) = prefixn(valid-group-nodes(gndx0), lax0).to-adr(lnum) + abs-log(nx1, gstore ↓), lnum ∈ ginval

Rule argument

spec: <toplevel> name: down-store-at seq: ⊦ n ∈ gstore → gstore ↓[n] = gstore[n] ↓ subst: [n, gstore] → [lnum, crash(gstore, gstore0, ginval, nx0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ∈ crash(gstore, gstore0, ginval, nx) ↔ n ∈ gstore ∨ n ∈ ginval; (ws) Used terms: lnum, crash(gstore, gstore0, ginval, nx0)