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Proof step 36 from proof btree-istore-reduce-refed (previous step)

# asar = # asar0, # zbrar = BRANCH_SIZE, adr ∈ as ↔ ¬ dirty0, union(asar0, 0, # asar0) = alladrs -- adr, union(asar, 0, # asar0) = as -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, # asar0, ri), iast0, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r => mkznode(rp, zbrar, false, dirty0, # asar0)) * P) )(zh0), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, # asar0 - 1), unmodified(adr, mkznode(rp, zbrar, false, dirty0, # asar0), iast0), adr ∈ alladrs, # asar0 ≤ BRANCH_SIZE, iast0 | as ⊆ iast1, as ⊆ dom iast0, as ⊆ alladrs, dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh), ∀ rp, rp0, adr, dirty, ri, iast0, as, alladrs, nstat, P, zh, iast1. ((btree(rp, rp0, adr, height0, dirty, ri, iast0, as, alladrs, nstat) * P))(zh) ∧ iast0 | as ⊆ iast1 ∧ as ⊆ dom iast0 ∧ as ⊆ alladrs → ((btree(rp, rp0, adr, height0, dirty, ri, iast1, as, alladrs, nstat) * P))(zh) ⊦ # asar0 = 0, # alladrs = 0, alladrs = ∅, (( r => mkznode(rp, zbrar, false, dirty0, # asar0) * bnode(r, zbrar, 0, # asar0, # asar0, height0 + 1, dirty0, ri, iast1, as -- adr, alladrs -- adr) * P) )(zh0)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 37:
# zbrar = BRANCH_SIZE, # asar = # asar0, # asar0 = m + 1, adr ∈ as ↔ ¬ dirty0, union(asar, 0, # asar0) = as -- adr, union(asar0, 0, # asar0) = alladrs -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, # asar0, ri), iast0, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r => mkznode(rp, zbrar, false, dirty0, # asar0)) * P) )(zh0), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, dirty0, # asar0), iast0), adr ∈ alladrs, # asar0 ≤ BRANCH_SIZE, as ⊆ alladrs, as ⊆ dom iast0, iast0 | as ⊆ iast1, dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh), ∀ rp, rp0, adr, dirty, ri, iast0, as, alladrs, nstat, P, zh, iast1. ((btree(rp, rp0, adr, height0, dirty, ri, iast0, as, alladrs, nstat) * P))(zh) ∧ iast0 | as ⊆ iast1 ∧ as ⊆ dom iast0 ∧ as ⊆ alladrs → ((btree(rp, rp0, adr, height0, dirty, ri, iast1, as, alladrs, nstat) * P))(zh) ⊦ # alladrs = 0, # asar0 = 0, alladrs = ∅, (( r => mkznode(rp, zbrar, false, dirty0, # asar0) * bnode(r, zbrar, 0, # asar0, # asar0, height0 + 1, dirty0, ri, iast1, as -- adr, alladrs -- adr) * P) )(zh0)

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