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Proof step 45 from proof nodes-append-subseteq (previous step)

# gndx = # lax, prefixn(valid-group-nodes(gndx), gndx + mkgnode(nd, m3, boolvar, false) ')[pos(@(m1, m2), lax + ladr ')] = mkgnode(nd1, m4, boolvar3, boolvar4), gndx[pos(@(m1, m2), lax)] = mkgnode(nd0, m, boolvar1, boolvar2), gstore[n] = groupblock(gndx, lax, n0), pos(@(m1, m2), lax) < valid-group-nodes(gndx), pos(@(m1, m2), lax) < # lax, @(m1, m2) ∈ prefixn(valid-group-nodes(gndx), lax), n ∈ gstore, @(m1, m2) ∈ prefixn(valid-group-nodes(gndx), lax + ladr '), n0 ≤ m0 ⊦ lax = [], nd0 = nd1, n ∈ ginval

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: list-prefixn, inst: group-store name: prefixn-append-left seq: n ≤ # gndx ⊦ prefixn(n, gndx + gndx0) = prefixn(n, gndx) subst: [gndx0, gndx, n] → [mkgnode(nd, m3, boolvar, false) ', gndx, valid-group-nodes(gndx)] flags: <all paths>

Additional information

Used simplifier rules: valid-group-nodes-ls: ⊦ n < valid-group-nodes(gndx) → n < # gndx;% ⊦ #(gndx + gndx0) = # gndx + # gndx0; (s)% ⊦ # gnd ' = 0 +1; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%# gndx ≤ m ⊦ prefixn(m, gndx) = gndx; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%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]% ⊦ m +1 = m + 1; (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 ↔ false); (s)%# lax ≤ m ⊦ prefixn(m, lax) = lax; (s)% ⊦ ladr ∈ lax + lax0 ↔ ladr ∈ lax ∨ ladr ∈ lax0; (s)% ⊦ ladr ∈ ladr0 ' ↔ ladr = ladr0; (s)% ⊦ ladr ∈ lax → pos(ladr, lax + lax0) = pos(ladr, lax); (s)% ⊦ ladr ∈ lax → pos(ladr, lax) < # lax; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ # gndx = 0 ↔ gndx = []; (s)% ⊦ ¬ n < n; (s) Used terms: mkgnode(nd, m3, boolvar, false) ', gndx, valid-group-nodes(gndx)