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Proof step 28 from proof range-put (previous step)

2 + m < nused, m + 1 < nused, zbrar[m].key < key, 2 + m < # zbrar, m + 1 < # zbrar, nused ≤ # zbrar ⊦ zbrar[m + 1].key < key, (key0 = key ∨ key0 ∈ ri ∧ zbrar[m].key < key0 ∧ ¬ zbrar[m + 1].key < key0 → key0 ∈ ((ri |> zbrar[m].key) |≤ zbrar[m + 1].key)[key, adr]) ∧ ( key0 ∈ ((ri |> zbrar[m].key) |≤ zbrar[m + 1].key)[key, adr] → ri[key, adr][key0] = ((ri |> zbrar[m].key) |≤ zbrar[m + 1].key)[key, adr][key0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ key ≠ key0 → ri[key0, adr][key] = ri[key]
Proof step 30:
m + 1 < # zbrar, 2 + m < # zbrar, zbrar[m].key < key, m + 1 < nused, 2 + m < nused, nused ≤ # zbrar ⊦ key0 = key, zbrar[m + 1].key < key, (key0 = key ∨ key0 ∈ ri ∧ zbrar[m].key < key0 ∧ ¬ zbrar[m + 1].key < key0 → key0 ∈ ((ri |> zbrar[m].key) |≤ zbrar[m + 1].key)[key, adr]) ∧ (key0 ∈ ((ri |> zbrar[m].key) |≤ zbrar[m + 1].key)[key, adr] → ri[key, adr][key0] = ((ri |> zbrar[m].key) |≤ zbrar[m + 1].key)[key0])

Rule argument

spec: store, inst: node-index name: At-other seq: ⊦ key ≠ key0 → ri[key0, adr][key] = ri[key] subst: [key, adr, key0, ri] → [key0, adr, key, (ri |> zbrar[m].key) |≤ zbrar[m + 1].key] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key ∈ ri[key, adr]; (s)% ⊦ ri[key, adr][key] = adr; (s) Used terms: key0, adr, key, (ri |> zbrar[m].key) |≤ zbrar[m + 1].key