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Proof step 27 from proof btree-clean-fwd (previous step)

# asar = # asar0, # zbrar0 = BRANCH_SIZE, union(asar0, 0, # asar0) = alladrs0 -- adr, union(asar, 0, # asar0) = alladrs0 -- adr, (( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(r2, zbrar0[n].child, zbrar0[n].adr, height, false, key-range(zbrar0, n, # asar0, ri0), iast, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * P0) )(zh), ((bnode(r, zbrar, 0, # asar0, # asar0, height0 + 1, false, ri1, iast, alladrs1 -- adr, alladrs1 -- adr) * P1))(zh3), ((r => mkznode(rp1, zbrar, false, false, # asar0)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, # asar0)))(zh5), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar0, 0, # asar0 - 1), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar0), iast), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar0), iast), adr ∈ alladrs0, adr ∈ alladrs1, adr ∈ iast, # asar0 ≤ BRANCH_SIZE, alladrs0 ⊆ dom iast, alladrs1 ⊆ dom iast, zh4 ⊥ zh3, zh5 ⊥ zh, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0), ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # asar0 = 0, # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, height = height0 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 28:
# zbrar0 = BRANCH_SIZE, # asar = # asar0, # asar0 = m + 1, union(asar, 0, # asar0) = alladrs0 -- adr, union(asar0, 0, # asar0) = alladrs0 -- adr, ((bnode(r, zbrar, 0, # asar0, # asar0, height0 + 1, false, ri1, iast, alladrs1 -- adr, alladrs1 -- adr) * P1))(zh3), (( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(r2, zbrar0[n].child, zbrar0[n].adr, height, false, key-range(zbrar0, n, # asar0, ri0), iast, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * P0) )(zh), ((r2 => mkznode(rp0, zbrar0, false, false, # asar0)))(zh5), ((r => mkznode(rp1, zbrar, false, false, # asar0)))(zh4), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar0), iast), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar0), iast), adr ∈ iast, adr ∈ alladrs1, adr ∈ alladrs0, # asar0 ≤ BRANCH_SIZE, alladrs1 ⊆ dom iast, alladrs0 ⊆ dom iast, zh5 ⊥ zh, zh4 ⊥ zh3, (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0), (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # alladrs1 = 0, # alladrs0 = 0, # asar0 = 0, alladrs0 = ∅, alladrs1 = ∅, height = height0 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

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