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Proof step 7 from proof valid-group-nodes-prefixn-append-nonend (previous step)

n ≤ # gndx + 1 ⊦ valid-group-nodes(firstn(n, gndx + mkgnode(nd, m, boolvar, false) ')) = valid-group-nodes((n ≤ # gndx ⊃ firstn(n, gndx);gndx))

User applied rule case distinction

And closed the goal!

Rule argument

pos: right 1

Additional information

Used simplifier rules: valid-group-nodes-rec: ⊦ valid-group-nodes(gndx + gnd ') = (gnd.end ⊃ # gndx + 1;valid-group-nodes(gndx));% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ 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%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ firstn(# gndx + n, gndx + gndx0) = gndx + firstn(n, gndx0); (s)% ⊦ firstn(1, gnd ') = gnd '; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ mkgnode(nd, m, boolvar, boolvar0).end ↔ boolvar0; (s)% ⊦ m ≤ # gndx → firstn(m, gndx + gndx0) = firstn(m, gndx); (s)