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Proof step 27 from proof btree-write-node (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 28:
# zbrar = BRANCH_SIZE, # asar = # asar0, # asar0 = m + 1, union(asar0, 0, # asar0) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, # asar0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r => mkznode(r0, zbrar, false, true, # asar0)) * P) )(zh), btree-sub-clean(r, # asar0, zh), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, m), adr0 ∈ alladrs, # asar0 ≤ BRANCH_SIZE, ∀ n. n < # asar0 → asar[n] ⊆ alladrs, ∀ n. n < # asar0 → asar[n] ⊆ dom iast ⊦ # alladrs = 0, # asar0 = 0, alladrs = ∅, adr0 ∈ union(asar, 0, # asar0), adr1 ∈ alladrs, adr1 ∈ iast, ∃ asar2, asar1. # asar2 = # asar1 ∧ disj(asar2, 0, # asar1) ∧ union(asar2, 0, # asar1) = union(asar, 0, # asar0) -- adr1 ∧ # asar1 = # asar0 ∧ 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)

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [# asar0, m]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # asar0, m