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Proof step 63 from proof prepost-gc-referenced-nodes (previous step)

# gndx0 = # lax1, # lax = # gndx2, prefixn(valid-group-nodes(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2), gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2)[# lax1] = mkgnode(nd0, m4, boolvar1, boolvar2), ri[nd.key] = @(m1, m2, m3), gstore2[m1] = groupblock(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, lax1 + @(m2, m3) ' + lax, m), inv(gstore2 \ ginval), inv-flashsize(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, lax1 + @(m2, m3) ' + lax), key-cons(nd.key, nd0), @(m2, m3) ∈ prefixn(valid-group-nodes(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2), lax1 + @(m2, m3) ' + lax), nd.key ∈ ri, m1 ∈ gstore2 ⊦ nd = nd0, dups(lax), dups(lax1), dups(lax1 + lax), @(m2, m3) ∈ lax, @(m2, m3) ∈ lax1, m1 ∈ ginval

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 64:
⊦ m < n ∧ m < # gndx → prefixn(n, gndx)[m] = gndx[m]
Proof step 65:
# lax = # gndx2, # gndx0 = # lax1, gstore2[m1] = groupblock(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, lax1 + @(m2, m3) ' + lax, m), ri[nd.key] = @(m1, m2, m3), (gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2)[# lax1] = mkgnode(nd0, m4, boolvar1, boolvar2), # lax1 < valid-group-nodes(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2), # lax1 < #(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2), inv(gstore2 \ ginval), inv-flashsize(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, lax1 + @(m2, m3) ' + lax), key-cons(nd.key, nd0), m1 ∈ gstore2, nd.key ∈ ri, @(m2, m3) ∈ prefixn(valid-group-nodes(gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2), lax1 + @(m2, m3) ' + lax) ⊦ nd = nd0, dups(lax1 + lax), dups(lax1), dups(lax), m1 ∈ ginval, @(m2, m3) ∈ lax1, @(m2, m3) ∈ lax

Rule argument

spec: list-prefixn, inst: group-store name: in-prefixn-at-smaller seq: ⊦ m < n ∧ m < # gndx → prefixn(n, gndx)[m] = gndx[m] subst: [m, gndx, n] → [# lax1, gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, valid-group-nodes (gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(gndx + gndx0) = # gndx + # gndx0; (s)% ⊦ #(gnd ' + gndx) = (# gndx)+1; (s)% ⊦ m +1 = m + 1; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ m < n + 1 ↔ ¬ n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ m + n < m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%n ≤ # gndx ⊦ prefixn(n, gndx + gndx0) = prefixn(n, gndx); (s)%n ≤ # lax ⊦ prefixn(n, lax + lax0) = prefixn(n, lax); (s)%a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list]% ⊦ ¬ ladr ∈ lax → ¬ ladr ∈ prefixn(n, lax); (s) Used terms: # lax1, gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, valid-group-nodes (gndx0 + mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2)