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Proof step 55 from proof btree-ri-unique (previous step)

# asar1 = (n0 + 1) + n, # asar0 = # asar1, # zbrar0 = BRANCH_SIZE, # asar = # asar1, # asar2 = # asar1, adr ∈ as0 ↔ adr ∈ as1, adr ∈ as1 ↔ ¬ dirty1, zbrar0[n] = mkzbranch(key, adr0, null), union(asar2, 0, # asar1) = alladrs0 -- adr, union(asar1, 0, # asar1) = as0 -- adr, union(asar, 0, # asar1) = as1 -- adr, union(asar0, 0, # asar1) = alladrs1 -- adr, ((btree(rp0, null, adr0, height0, dirty1, key-range(zbrar0, n, # asar1, ri1), iast, asar[n], asar0[n], ok) * P2))(zh), ((btree(rp0, null, adr0, height0, dirty1, key-range(zbrar0, n, # asar1, ri0), iast, asar1[n], asar2[n], ok) * P))(zh), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, n + 1 < BRANCH_SIZE, # asar1 < BRANCH_SIZE, disj(asar2, 0, # asar1), disj(asar1, 0, # asar1), disj(asar, 0, # asar1), disj(asar0, 0, # asar1), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar1), iast), adr ∈ alladrs1, adr ∈ alladrs0, asar[n] ⊆ asar0[n], asar[n] ⊆ dom iast, as1 ⊆ dom iast, as1 ⊆ alladrs1, as0 ⊆ dom iast, as0 ⊆ alladrs0, asar1[n] ⊆ asar2[n], asar1[n] ⊆ dom iast, dirty1 → dirty ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar0, n, # asar1, ri0) = key-range(zbrar0, n, # asar1, ri1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ r = null ∨ ¬ zh0[r].dirty → ( ((btree(rp, r, adr, height, dirty, ri, iast, as, alladrs, nstat) * P))(zh0) ↔ ((btree(rp, r, adr, height, false, ri, iast, as, alladrs, nstat) * P))(zh0))
Proof step 57:
# asar2 = # asar1, # asar = # asar1, # zbrar0 = BRANCH_SIZE, # asar0 = # asar1, # asar1 = (n0 + 1) + n, adr ∈ as1 ↔ ¬ dirty1, adr ∈ as0 ↔ adr ∈ as1, zbrar0[n] = mkzbranch(key, adr0, null), union(asar0, 0, # asar1) = alladrs1 -- adr, union(asar, 0, # asar1) = as1 -- adr, union(asar1, 0, # asar1) = as0 -- adr, union(asar2, 0, # asar1) = alladrs0 -- adr, ((btree(rp0, null, adr0, height0, dirty1, key-range(zbrar0, n, # asar1, ri0), iast, asar1[n], asar2[n], ok) * P))(zh), ((btree(rp0, null, adr0, height0, false, key-range(zbrar0, n, # asar1, ri1), iast, asar[n], asar0[n], ok) * P2))(zh), # asar1 < BRANCH_SIZE, n + 1 < BRANCH_SIZE, n0 + n < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, # asar1), disj(asar, 0, # asar1), disj(asar1, 0, # asar1), disj(asar2, 0, # asar1), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar1), iast), adr ∈ alladrs0, adr ∈ alladrs1, asar1[n] ⊆ dom iast, asar1[n] ⊆ asar2[n], as0 ⊆ alladrs0, as0 ⊆ dom iast, as1 ⊆ alladrs1, as1 ⊆ dom iast, asar[n] ⊆ dom iast, asar[n] ⊆ asar0[n], dirty1 → dirty, null = null ∨ ¬ zh[null].dirty ⊦ # alladrs1 = 0, # alladrs0 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar0, n, # asar1, ri0) = key-range(zbrar0, n, # asar1, ri1)

Rule argument

spec: <toplevel> name: btree-null-or-clean-below seq: ⊦ r = null ∨ ¬ zh0[r].dirty → ( (( btree (rp, r, adr, height, dirty, ri, iast, as, alladrs, nstat) * P) )(zh0) ↔ (( btree (rp, r, adr, height, false, ri, iast, as, alladrs, nstat) * P) )(zh0)) subst: [zh0, P, nstat, alladrs, as, iast, ri, dirty, height, adr, r, rp] → [zh, P2, ok, asar0[n], asar[n], iast, key-range(zbrar0, n, # asar1, ri1), dirty1, height0, adr0, null, rp0] flags: <all paths>

Additional information

Used terms: zh, P2, ok, asar0[n], asar[n], iast, key-range(zbrar0, n, # asar1, ri1), dirty1, height0, adr0, null, rp0