[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 66 from proof prepost-gc-referenced-nodes (previous step)

# gndx0 = # lax1, # lax = # 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), # lax1 < valid-group-nodes(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), @(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

Rule argument

spec: list-del, inst: group-store name: get-append-grt seq: ⊦ ¬ n < # gndx → (gndx + gndx0)[n] = gndx0[n - # gndx] subst: [n, gndx0, gndx] → [# lax1, mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, gndx0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m - m = 0; (s)% ⊦ (gnd ' + gndx)[0] = gnd; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]%⊦ mkgnode(nd, m, boolvar, boolvar0) = mkgnode(nd0, m0, boolvar1, boolvar2) ↔ nd = nd0 ∧ m = m0 ∧ (boolvar ↔ boolvar1) ∧ (boolvar0 ↔ boolvar2); (ws)% ⊦ n < # gndx → (gndx + gndx0)[n] = gndx[n]; (s)% ⊦ #(lax + lax0) = # lax + # lax0; (s)% ⊦ # ladr ' = 0 +1; (s)%a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list]% ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic% ⊦ # gndx = 0 ↔ gndx = []; (s)% ⊦ ¬ n < n; (s)%n ≤ # lax ⊦ prefixn(n, lax + lax0) = prefixn(n, lax); (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)%# lax ≤ m ⊦ prefixn(m, lax) = lax; (s)% ⊦ ladr ∈ lax + lax0 ↔ ladr ∈ lax ∨ ladr ∈ lax0; (s)% ⊦ ladr ∈ ladr0 ' ↔ ladr = ladr0; (s)%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]% ⊦ # lax = 0 ↔ lax = []; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m < n → ¬ n < m; (s) Used terms: # lax1, mkgnode(nd, m0, boolvar, boolvar0) ' + gndx2, gndx0