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Proof step 28 from proof btree-ri-unique (previous step)

# asar = m + 1, # zbrar0 = BRANCH_SIZE, # asar0 = m + 1, adr ∈ as1 ↔ ¬ dirty1, adr ∈ as0 ↔ adr ∈ as1, union(asar0, 0, m + 1) = alladrs1 -- adr, union(asar, 0, m + 1) = as1 -- adr, (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1)) * P1) )(zh), (( rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1) * bnode(rp0, zbrar0, 0, m + 1, m + 1, height0 + 1, dirty1, ri0, iast, as0 -- adr, alladrs0 -- adr) * P0) )(zh), m + 1 < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, m + 1), iast), adr ∈ alladrs0, adr ∈ alladrs1, as0 ⊆ alladrs0, as0 ⊆ dom iast, as1 ⊆ alladrs1, as1 ⊆ dom iast, dirty1 → dirty, ∀ rp, rp0, adr, dirty, ri0, iast, as0, alladrs0, P0, zh, ri1, as1, alladrs1, P1. ((btree(rp, rp0, adr, height0, dirty, ri0, iast, as0, alladrs0, ok) * P0))(zh) ∧ ((btree(rp, rp0, adr, height0, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh) ∧ as0 ⊆ alladrs0 ∧ as0 ⊆ dom iast ∧ as1 ⊆ alladrs1 ∧ as1 ⊆ dom iast → ri0 = ri1 ∨ rp0 = null ⊦ # alladrs1 = 0, # alladrs0 = 0, alladrs0 = ∅, alladrs1 = ∅, ri0 = ri1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ bnode(rp, zbrar, m, len, nused, height + 1, dirty, ri, iast, as, alladrs) = ( ⌜m + len ≤ nused ∧ nused ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜nused ≠ 0 ∧ sorted(zbrar, 0, nused - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = nused ∧ disj(asar, 0, nused) ∧ union(asar, m, len) = as ∧ # asar0 = nused ∧ disj(asar0, 0, nused) ∧ union(asar0, m, len) = alladrs ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(rp, zbrar[n].child, zbrar[n].adr, height, dirty, key-range(zbrar, n, nused, ri), iast, asar[n], asar0[n], ok) )⊗_ (m, len ))))
Proof step 30:
# asar0 = m + 1, # zbrar0 = BRANCH_SIZE, # asar = m + 1, adr ∈ as0 ↔ adr ∈ as1, adr ∈ as1 ↔ ¬ dirty1, union(asar, 0, m + 1) = as1 -- adr, union(asar0, 0, m + 1) = alladrs1 -- adr, (( rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1) * ( ⌜0 + m + 1 ≤ m + 1 ∧ m + 1 ≤ # zbrar0 ∧ # zbrar0 = BRANCH_SIZE⌝ ∧ ⌜m + 1 ≠ 0 ∧ sorted(zbrar0, 0, m + 1 - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = m + 1 ∧ disj(asar, 0, m + 1) ∧ union(asar, 0, m + 1) = as0 -- adr ∧ # asar0 = m + 1 ∧ disj(asar0, 0, m + 1) ∧ union(asar0, 0, m + 1) = alladrs0 -- adr ⌝ ∧ (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri0), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 )))) * P0) )(zh), (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1)) * P1) )(zh), m + 1 < BRANCH_SIZE, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, m + 1), iast), adr ∈ alladrs1, adr ∈ alladrs0, as1 ⊆ dom iast, as1 ⊆ alladrs1, as0 ⊆ dom iast, as0 ⊆ alladrs0, dirty1 → dirty, ∀ rp, rp0, adr, dirty, ri0, iast, as0, alladrs0, P0, zh, ri1, as1, alladrs1, P1. ((btree(rp, rp0, adr, height0, dirty, ri0, iast, as0, alladrs0, ok) * P0))(zh) ∧ ((btree(rp, rp0, adr, height0, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh) ∧ as0 ⊆ alladrs0 ∧ as0 ⊆ dom iast ∧ as1 ⊆ alladrs1 ∧ as1 ⊆ dom iast → ri0 = ri1 ∨ rp0 = null ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, ri0 = ri1

Rule argument

spec: <toplevel> name: bnode seq: ⊦ bnode (rp, zbrar, m, len, nused, height + 1, dirty, ri, iast, as, alladrs) = ( ⌜ m + len ≤ nused ∧ nused ≤ # zbrar ∧ # zbrar = BRANCH_SIZE ⌝ ∧ ⌜nused ≠ 0 ∧ sorted(zbrar, 0, nused - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = nused ∧ disj(asar, 0, nused) ∧ union(asar, m, len) = as ∧ # asar0 = nused ∧ disj(asar0, 0, nused) ∧ union(asar0, m, len) = alladrs ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (rp, zbrar[n].child, zbrar[n].adr, height, dirty, key-range(zbrar, n, nused, ri), iast, asar[n], asar0[n], ok) )⊗_ (m, len )))) subst: [alladrs, as, iast, ri, dirty, height, nused, len, m, zbrar, rp] → [alladrs0 -- adr, as0 -- adr, iast, ri0, dirty1, height0, m + 1, m + 1, 0, zbrar0, rp0] flags: <all paths>

Additional information

Used terms: alladrs0 -- adr, as0 -- adr, iast, ri0, dirty1, height0, m + 1, m + 1, 0, zbrar0, rp0