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Proof step 47 from proof btree-adrs (previous step)

# zbrar = BRANCH_SIZE, # asar = # asar0, # asar0 = (n0 + n) + 1, zbrar[n] = mkzbranch(key, adr, r0), union(asar0, 0, # asar0) = alladrs -- adr0, (( ( ( btree(r, r0, adr, height0, false, key-range(zbrar, n, # asar0, fi), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar0, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar0, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * P) )(zh1), zbrar[n].branch?, n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, # asar0)))(zh2), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, n0 + n), unmodified(adr0, mkznode(rp, zbrar, false, false, # asar0), iast), adr0 ∈ iast, adr0 ∈ alladrs, asar[n] ⊆ asar0[n], zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, # asar0), asar[n] ⊆ dom iast, alladrs ⊆ dom iast, asar0[n] ⊆ dom iast

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 48:
# zbrar = BRANCH_SIZE, # asar = # asar0, # asar0 = (n0 + n) + 1, zbrar[n] = mkzbranch(key, adr, r0), union(asar0, 0, # asar0) = alladrs -- adr0, (( ( ( btree(r, r0, adr, height0, false, key-range(zbrar, n, # asar0, fi), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar0, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar0, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * P) )(zh1), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, # asar0)))(zh2), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, n0 + n), unmodified(adr0, mkznode(rp, zbrar, false, false, # asar0), iast), adr0 ∈ iast, adr0 ∈ alladrs, asar[n] ⊆ asar0[n], zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, # asar0), asar[n] ⊆ dom iast, alladrs ⊆ dom iast, asar0[n] ⊆ dom iast

Rule argument

Additional information

Used simplifier rules: nat-le-eq: ⊦ n = n0 + n1 → (n ≤ n0 ↔ n1 = 0 ∧ n = n0);%⊦ mkzbranch(key, adr, r) = mkzbranch(key0, adr0, r0) ↔ key = key0 ∧ adr = adr0 ∧ r = r0; (ws)%unmodified-clean-fwd-zbrar: n < nused, nused ≤ BRANCH_SIZE, # zbrar0 = # zbrar1, # zbrar0 = BRANCH_SIZE ⊦ zbrar0[n] = mkzbranch(key, adr0, r2) ∧ unmodified(adr, mkznode(rp0, zbrar0, bv, false, nused), iast) ∧ unmodified(adr, mkznode(rp1, zbrar1, bv, false, nused), iast) → (∃ r0. zbrar1[n] = mkzbranch(key, adr0, r0)); (forward) from specification zbranch-array-ops% ⊦ nsize(m, nstat) → ¬ BRANCH_SIZE < m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m + 1 ≠ 0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ mkzbranch(key, adr, r).branch?; (s)