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Proof step 29 from proof btree-write-node (previous step)

# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, union(asar0, 0, m + 1) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(r0, zbrar, false, true, m + 1)) * P) )(zh), m < BRANCH_SIZE, btree-sub-clean(r, m + 1, zh), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), adr0 ∈ alladrs, ∀ n. ¬ m < n → asar[n] ⊆ alladrs, ∀ n. ¬ m < n → asar[n] ⊆ dom iast ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, m + 1), adr1 ∈ alladrs, adr1 ∈ iast, ∃ asar2, asar1. # asar2 = # asar1 ∧ disj(asar2, 0, # asar1) ∧ union(asar2, 0, # asar1) = union(asar, 0, m + 1) -- adr1 ∧ # asar1 = m + 1 ∧ disj(asar1, 0, # asar1) ∧ union(asar1, 0, # asar1) = alladrs -- adr0 -- adr1 ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, # asar1))], asar2[n], asar1[n], ok) )⊗_ (0, # asar1 ) * r =>_) * P) )(zh)

User applied rule exists right

And obtained the following new goals

Proof step 30:
# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, union(asar0, 0, m + 1) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(r0, zbrar, false, true, m + 1)) * P) )(zh), m < BRANCH_SIZE, btree-sub-clean(r, m + 1, zh), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), adr0 ∈ alladrs, ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ union(asar, 0, m + 1), # asar = # asar0 ∧ disj(asar, 0, # asar0) ∧ union(asar, 0, # asar0) = union(asar, 0, m + 1) -- adr1 ∧ # asar0 = m + 1 ∧ disj(asar0, 0, # asar0) ∧ union(asar0, 0, # asar0) = alladrs -- adr0 -- adr1 ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar0, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, # asar0))], asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r =>_) * P) )(zh)

Rule argument

pos: right 6 terms: asar, asar0 abort: F discard: T computed: T

Additional information

Used terms: asar, asar0