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Proof step 39 from proof btree-ri-unique (previous step)

# asar2 = (n0 + n) + 1, # asar1 = # asar2, # asar0 = # asar2, # zbrar0 = BRANCH_SIZE, # asar = # asar2, adr ∈ as0 ↔ adr ∈ as1, adr ∈ as1 ↔ ¬ dirty1, union(asar2, 0, # asar2) = alladrs0 -- adr, union(asar1, 0, # asar2) = as0 -- adr, union(asar, 0, # asar2) = as1 -- adr, union(asar0, 0, # asar2) = alladrs1 -- adr, (( ( ( ( btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, # asar2, ri1), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree (rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P1) )(zh), (( ( (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok) )⊗_ (0, # asar2 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P0) )(zh), zbrar0[n].branch?, n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, # asar2 < BRANCH_SIZE, disj(asar2, 0, # asar2), disj(asar1, 0, # asar2), disj(asar, 0, # asar2), disj(asar0, 0, # asar2), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar2), iast), adr ∈ alladrs1, adr ∈ alladrs0, asar[n] ⊆ asar0[n], asar[n] ⊆ dom iast, 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 = ∅, key-range(zbrar0, n, # asar2, ri0) = key-range(zbrar0, n, # asar2, ri1)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 40:
# asar = # asar2, # zbrar0 = BRANCH_SIZE, # asar0 = # asar2, # asar1 = # asar2, # asar2 = (n0 + n) + 1, adr ∈ as1 ↔ ¬ dirty1, adr ∈ as0 ↔ adr ∈ as1, zbrar0[n] = mkzbranch(key, adr0, r), union(asar0, 0, # asar2) = alladrs1 -- adr, union(asar, 0, # asar2) = as1 -- adr, union(asar1, 0, # asar2) = as0 -- adr, union(asar2, 0, # asar2) = alladrs0 -- adr, (( ( (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok) )⊗_ (0, # asar2 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P0) )(zh), (( ( ( ( btree(rp0, r, adr0, height0, dirty1, key-range(zbrar0, n, # asar2, ri1), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree (rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P1) )(zh), zbrar0[n].branch?, # asar2 < BRANCH_SIZE, n0 + n < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, # asar2), disj(asar, 0, # asar2), disj(asar1, 0, # asar2), disj(asar2, 0, # asar2), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar2), iast), adr ∈ alladrs0, adr ∈ alladrs1, as0 ⊆ alladrs0, as0 ⊆ dom iast, as1 ⊆ alladrs1, as1 ⊆ dom iast, asar[n] ⊆ dom iast, asar[n] ⊆ asar0[n], 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 = ∅, key-range(zbrar0, n, # asar2, ri0) = key-range(zbrar0, n, # asar2, ri1)

Rule argument

spec: zbranch name: elim-01 seq: ⊦ zbr.branch? → ( (key = zbr.key ∧ adr = zbr.adr) ∧ r = zbr.child ↔ zbr = mkzbranch(key, adr, r)) subst: [zbr, key, adr, r] → [zbrar0[n], key, adr0, r]

Additional information

Used simplifier rules: ⊦ zbr.branch? → ( (key = zbr.key ∧ adr = zbr.adr) ∧ r = zbr.child ↔ zbr = mkzbranch(key, adr, r)); (elim) Used terms: zbrar0[n], key, adr0, r