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Proof step 20 from proof range-put (previous step)

nused0 < # zbrar ⊦ zbrar[0].key < key, (key0 ∈ key-range(zbrar, 0, nused0 + 1, ri[key, adr]) ↔ key0 = key ∨ key0 ∈ key-range(zbrar, 0, nused0 + 1, ri)) ∧ (key0 ∈ key-range(zbrar, 0, nused0 + 1, ri[key, adr]) → ri[key, adr][key0] = key-range(zbrar, 0, nused0 + 1, ri)[key, adr][key0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ nused ≠ 0 ∧ nused < # zbrar → key-range(zbrar, 0, nused + 1, ri) = ri |≤ zbrar[0].key
Proof step 22:
nused0 < # zbrar ⊦ nused0 = 0, zbrar[0].key < key, (key0 ∈ ri[key, adr] |≤ zbrar[0].key ↔ key0 = key ∨ key0 ∈ key-range(zbrar, 0, nused0 + 1, ri)) ∧ (key0 ∈ ri[key, adr] |≤ zbrar[0].key → ri[key, adr][key0] = key-range(zbrar, 0, nused0 + 1, ri)[key, adr][key0])

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[key, adr], nused0, zbrar] flags: <all paths>

Additional information

Used simplifier rules: range-one: ⊦ key-range(zbrar, 0, 1, ri) = ri;%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ m < n → n ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ key ≠ key0 → (key ∈ ri[key0, adr] ↔ key ∈ ri); (s)% ⊦ key ∈ ri[key0, adr] ↔ key = key0 ∨ key ∈ ri; (ws) Used terms: ri[key, adr], nused0, zbrar