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Proof step 10 from proof keep-range (previous step)

zbr.key = zbrar[m].key, m < # zbrar, nused0 < # zbrar ⊦ key-range(zbrar[m, zbr], 0, nused0 + 1, ri) = key-range(zbrar, 0, nused0 + 1, ri)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ nused ≠ 0 ∧ nused < # zbrar → key-range(zbrar, 0, nused + 1, ri) = ri |≤ zbrar[0].key
Proof step 12:
zbr.key = zbrar[m].key, nused0 < # zbrar, m < # zbrar, nused0 < # zbrar[m, zbr] ⊦ nused0 = 0, ri |≤ zbrar[m, zbr][0].key = key-range(zbrar, 0, nused0 + 1, ri)

Rule argument

spec: <toplevel> name: range-first seq: ⊦ nused ≠ 0 ∧ nused < # zbrar → key-range(zbrar, 0, nused + 1, ri) = ri |≤ zbrar[0].key subst: [ri, nused, zbrar] → [ri, nused0, zbrar[m, zbr]] flags: <all paths>

Additional information

Used simplifier rules: range-one: ⊦ key-range(zbrar, 0, 1, ri) = ri;% ⊦ # zbrar[n, zbr] = # zbrar; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: ri, nused0, zbrar[m, zbr]