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Proof step 47 from proof btree-clean-alladrs (previous step)

# asar0 = # asar, # asar = (n0 + n) + 1, # zbrar = BRANCH_SIZE, zbrar[n] = mkzbranch(key, adr1, r1), (( ( ( ( btree(r0, r1, adr1, height0, false, key-range(zbrar, n, # asar, ri), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * r0 => mkznode(rp, zbrar, false, false, # asar)) * P) )(zh0), n0 + n < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, # asar), disj(asar, 0, # asar), nsize(# asar, nstat), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar, false, false, # asar), iast), adr0 ∈ asar0[n], adr ∈ iast, asar[n] ⊆ dom iast, asar[n] ⊆ asar0[n], (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ dom iast, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ union(asar0, 0, # asar) ++ adr, ∀ rp, r, adr, ri, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, false, ri, iast, as, alladrs, nstat) * P))(zh) → as = alladrs ⊦ # asar0[n] = 0, # union(asar0, 0, # asar) = 0, asar0[n] = ∅, union(asar0, 0, # asar) = ∅, adr ∈ union(asar, 0, # asar), adr ∈ union(asar0, 0, # asar), adr0 ∈ union(asar, 0, # asar), adr0 ∈ asar[n]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ ((btree(rp, r, adr, height, dirty, ri, iast, as, alladrs, ok) * P))(zh) ↔ (∃ P0. P0 ≡ P ∧ ((btree(rp, r, adr, height, dirty, ri, iast, as, alladrs, ok) * P0))(zh))
Proof step 49:
# zbrar = BRANCH_SIZE, # asar = (n0 + n) + 1, # asar0 = # asar, zbrar[n] = mkzbranch(key, adr1, r1), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, disj(asar, 0, # asar), disj(asar0, 0, # asar), nsize(# asar, nstat), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar, false, false, # asar), iast), adr ∈ iast, adr0 ∈ asar0[n], asar[n] ⊆ asar0[n], asar[n] ⊆ dom iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, ri, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, false, ri, iast, as, alladrs, nstat) * P))(zh) → as = alladrs, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ union(asar0, 0, # asar) ++ adr, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ dom iast, ∃ P0. P0 ≡ P * r0 => mkznode(rp, zbrar, false, false, # asar) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 ) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n ) ∧ ((btree(r0, r1, adr1, height0, false, key-range(zbrar, n, # asar, ri), iast, asar[n], asar0[n], ok) * P0))(zh0) ⊦ # union(asar0, 0, # asar) = 0, # asar0[n] = 0, asar0[n] = ∅, union(asar0, 0, # asar) = ∅, adr0 ∈ asar[n], adr0 ∈ union(asar, 0, # asar), adr ∈ union(asar0, 0, # asar), adr ∈ union(asar, 0, # asar)

Rule argument

spec: <toplevel> name: btree-remove-context seq: ⊦ (( btree (rp, r, adr, height, dirty, ri, iast, as, alladrs, ok) * P) )(zh) ↔ (∃ P0. P0 ≡ P ∧ (( btree (rp, r, adr, height, dirty, ri, iast, as, alladrs, ok) * P0) )(zh)) subst: [zh, P, alladrs, as, iast, ri, dirty, height, adr, r, rp] → [zh0, P * r0 => mkznode(rp, zbrar, false, false, # asar) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 ) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n ), asar0[n], asar[n], iast, key-range(zbrar, n, # asar, ri), false, height0, adr1, r1, r0] flags: <all paths>

Additional information

Used simplifier rules: sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap] Used terms: zh0, P * r0 => mkznode(rp, zbrar, false, false, # asar) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 ) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n ), asar0[n], asar[n], iast, key-range(zbrar, n, # asar, ri), false, height0, adr1, r1, r0