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Proof step 80 from proof btree-clean-fwd-strong (previous step)

# asar2 = m + 1, # asar1 = m + 1, # zbrar = BRANCH_SIZE, # asar0 = m + 1, # asar = m + 1, # zbrar0 = BRANCH_SIZE, union(asar2, 0, m + 1) = alladrs1 -- adr, union(asar1, 0, m + 1) = alladrs1 -- adr, union(asar0, 0, m + 1) = alladrs0 -- adr, union(asar, 0, m + 1) = alladrs0 -- adr, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast1, asar1[n], asar2[n], ok) )⊗_ (0, m + 1 ) * P1) )(zh3), (( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(r2, zbrar0[n].child, zbrar0[n].adr, height, false, key-range(zbrar0, n, m + 1, ri0), iast0, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P0) )(zh), m < BRANCH_SIZE, ((r => mkznode(rp1, zbrar, false, false, m + 1)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, m + 1)))(zh5), disj(asar2, 0, m + 1), disj(asar1, 0, m + 1), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp0, zbrar0, false, false, m + 1), iast0), unmodified(adr, mkznode(rp1, zbrar, false, false, m + 1), iast1), adr ∈ alladrs0, adr ∈ iast0, adr ∈ alladrs1, adr ∈ iast1, iast0 ⊆ iast1, alladrs0 ⊆ dom iast0, alladrs1 ⊆ dom iast1, zh4 ⊥ zh3, zh5 ⊥ zh, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0), ∀ rp1, r1, adr, height1, ri1, iast1, alladrs1, nstat, P1, zh1, rp0, r0, ri0, iast0, alladrs0, P0, zh0. ((btree(rp1, r1, adr, height1, false, ri1, iast1, alladrs1, alladrs1, nstat) * P1))(zh1) ∧ ((btree(rp0, r0, adr, height, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0) ∧ alladrs1 ⊆ dom iast1 ∧ alladrs0 ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs0 = alladrs1, alladrs1 = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 81:
⊦ adr ∈ as → (as -- adr = as0 ↔ as = as0 ++ adr ∧ ¬ adr ∈ as0)
Proof step 82:
# zbrar0 = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, # zbrar = BRANCH_SIZE, # asar1 = m + 1, # asar2 = m + 1, alladrs1 = union(asar2, 0, m + 1) ++ adr, union(asar, 0, m + 1) = alladrs0 -- adr, union(asar0, 0, m + 1) = alladrs0 -- adr, union(asar1, 0, m + 1) = alladrs1 -- adr, (( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(r2, zbrar0[n].child, zbrar0[n].adr, height, false, key-range(zbrar0, n, m + 1, ri0), iast0, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P0) )(zh), (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast1, asar1[n], asar2[n], ok) )⊗_ (0, m + 1 ) * P1) )(zh3), m < BRANCH_SIZE, ((r2 => mkznode(rp0, zbrar0, false, false, m + 1)))(zh5), ((r => mkznode(rp1, zbrar, false, false, m + 1)))(zh4), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), disj(asar1, 0, m + 1), disj(asar2, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar0, 0, m), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp1, zbrar, false, false, m + 1), iast1), unmodified(adr, mkznode(rp0, zbrar0, false, false, m + 1), iast0), adr ∈ iast1, adr ∈ iast0, adr ∈ alladrs0, adr ∈ alladrs1, alladrs1 ⊆ dom iast1, alladrs0 ⊆ dom iast0, iast0 ⊆ iast1, zh5 ⊥ zh, zh4 ⊥ zh3, (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0), (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), ∀ rp1, r1, adr, height1, ri1, iast1, alladrs1, nstat, P1, zh1, rp0, r0, ri0, iast0, alladrs0, P0, zh0. ((btree(rp1, r1, adr, height1, false, ri1, iast1, alladrs1, alladrs1, nstat) * P1))(zh1) ∧ ((btree(rp0, r0, adr, height, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0) ∧ alladrs1 ⊆ dom iast1 ∧ alladrs0 ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # alladrs1 = 0, # alladrs0 = 0, alladrs0 = alladrs1, alladrs0 = ∅, alladrs1 = ∅, adr ∈ union(asar2, 0, m + 1)

Rule argument

spec: set-union, inst: node-store name: delete-shift seq: ⊦ adr ∈ as → (as -- adr = as0 ↔ as = as0 ++ adr ∧ ¬ adr ∈ as0) subst: [adr, as, as0] → [adr, alladrs1, union(asar2, 0, m + 1)] flags: <all paths>

Additional information

Used terms: adr, alladrs1, union(asar2, 0, m + 1)