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Proof step 43 from proof inv-commit-rec-ind (previous step)

# asar = m1 + 1, # zbrar0 = BRANCH_SIZE, # asar0 = m1 + 1, zh0[zbrar0[n0].child] = mkznode(r3, zbrar1, boolvar2, true, m2), istore2[adr3] = istore[adr3], adr = zbrar0[n0].adr, err = ESUCCESS, pstate = index(m3), remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, union(asar0, 0, m1 + 1) = alladrs -- adr3, elems nx ∩ ginval1 = ∅, as ∩ as0 = ∅, union(asar, 0, m1 + 1) ∩ as0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, # alladrs ≠ 0, zbrar0[n0].child ≠ null, alladrs ≠ ∅, (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, m1 + 1, ri), nodes(istore) | union(asar, 0, m1 + 1), asar[n], asar0[n], ok) )⊗_ (0, m1 + 1 ) * r0 => mkznode(rp, zbrar0, false, true, m1 + 1)) * P) )(zh0), ((btree(rp, r0, adr3, height1 + 1, true, ri, nodes(istore) | union(asar, 0, m1 + 1), union(asar, 0, m1 + 1), alladrs, ok) * P))(zh0), n0 < BRANCH_SIZE, m1 + 1 < BRANCH_SIZE, n0 + 1 < BRANCH_SIZE, btree-sub-clean(r0, n0, zh0), disj(asar0, 0, m1 + 1), disj(asar, 0, m1 + 1), inv(istore0), inv(istore2), inv(gstore1), inv(gstore2 \ ginval1), inv(istore), refsize-cons(istore2, as0 ∪ as), refsize-cons(istore, as0 ∪ union(asar, 0, m1 + 1)), sorted(zbrar0, 0, m1), adr3 ∈ istore, adr3 ∈ alladrs, m ∈ istore, m3 ∈ istore, m ∈ istore2, r0 ∈ zh1, adr3 ∈ istore2, n0 ≤ m1, as0 ⊆ dom nodes(istore2), alladrs ⊆ dom nodes(istore2), as ⊆ alladrs, as ⊆ dom nodes(istore2), union(asar, 0, m1 + 1) ∩ dom nodes(istore2) ⊆ as, union(asar, 0, m1 + 1) ∩ dom nodes(istore2) ⊆ alladrs, istore0 ⊑ istore, istore0 ⊑ istore2, istore2 ⊑ istore, ¬ dups(nx), ¬ adr3 ∈ as, ¬ adr3 ∈ union(asar, 0, m1 + 1), ∀ n. ¬ m1 < n → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, m1 + 1), ∀ n. ¬ m1 < n → asar[n] ⊆ alladrs, ∀ n. ¬ m1 < n → asar[n] ⊆ dom nodes(istore), ∀ n. ¬ m1 < n → asar[n] ⊆ dom nodes(istore2), ∀ height0, zh, r0, r, zbrar, boolvar, m0, adr0, adr1, istore1, istore, pstate, m, istore0, gstore0, nx, ginval, as, as0, gstore, rp, dirty, ri, alladrs, P, adr, ino, ns, err, bv. height0 < height1 +1 → zh[r0] = mkznode(r, zbrar, boolvar, true, m0) ∧ adr0 = adr1 ∧ istore1 = istore ∧ pstate = index(m) ∧ remove-refsize0(istore0) = istore0 ∧ remove-refsize0(gstore0) = gstore0 ∧ elems nx ∩ ginval = ∅ ∧ as ∩ as0 = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ ((btree(rp, r0, adr1, height0, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh) ∧ inv(istore0) ∧ inv(istore) ∧ inv(gstore0) ∧ inv(gstore \ ginval) ∧ refsize-cons(istore, as0 ∪ as) ∧ m ∈ istore ∧ r0 ∈ zh ∧ as0 ⊆ dom nodes(istore) ∧ alladrs ⊆ dom nodes(istore) ∧ as ⊆ alladrs ∧ as ⊆ dom nodes(istore) ∧ istore0 ⊑ istore ∧ ¬ dups(nx) ∧ ¬ adr1 ∈ as ∧ (height0 = 0 → as = ∅) → ⟪btree_commit_rec#(r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → bv) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ r0 ∈ zh ∧ (err = ESUCCESS → ¬ zh[r0].dirty ∧ pstate.index?) ∧ (∃ as1, alladrs1. refsize-cons(istore, as0 ∪ as1) ∧ as1 ∩ dom nodes(istore1) ⊆ as ∧ as1 ∩ as0 = ∅ ∧ alladrs1 ⊆ alladrs ∪ as1 ∧ ((btree(rp, r0, (err = ESUCCESS ⊃ adr1;adr0), height0, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh))) ⊦ ⟪btree_commit_rec#(zbrar0[n0].child, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr, err, zh0, istore, pstate, syn); n := n0; if err = ESUCCESS then zh0[r0].zbranches[n].adr := adr; n := n + 1⟫ ( commit-inv (n, pstate, istore2, istore0, nx, gstore2, gstore1, ginval1, syn, err, rp, r0, adr3, height1 + 1, true, ri, istore, as, as0, alladrs, P, zh0) ∧ ¬ m1 < (zh0[r0].usedsize - n) + n0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1)
Proof step 45:
# asar0 = m1 + 1, # zbrar0 = BRANCH_SIZE, # asar = m1 + 1, istore2[adr3] = istore[adr3], zh0[zbrar0[n0].child] = mkznode(r3, zbrar1, boolvar2, true, m2), adr = zbrar0[n0].adr, err = ESUCCESS, pstate = index(m3), remove-refsize0(gstore1) = gstore1, remove-refsize0(istore0) = istore0, union(asar0, 0, m1 + 1) = alladrs -- adr3, union(asar, 0, m1 + 1) ∩ as0 = ∅, as ∩ as0 = ∅, elems nx ∩ ginval1 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, ((btree(rp, r0, adr3, height1 + 1, true, ri, nodes(istore) | union(asar, 0, m1 + 1), union(asar, 0, m1 + 1), alladrs, ok) * P))(zh0), (( ( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, m1 + 1, ri), nodes(istore) | union(asar, 0, m1 + 1), asar[n], asar0[n], ok) )⊗_ (0, n0 ) * ( ⌜zbrar0[0 + n0].branch?⌝ ∧ btree (r0, zbrar0[0 + n0].child, zbrar0[0 + n0].adr, height1, true, key-range(zbrar0, 0 + n0, m1 + 1, ri), nodes(istore) | union(asar, 0, m1 + 1), asar[0 + n0], asar0[0 + n0], ok)) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, m1 + 1, ri), nodes(istore) | union(asar, 0, m1 + 1), asar[n], asar0[n], ok) )⊗_ (0 + n0 + 1, m1 + 1 - n0 - 1 )) * r0 => mkznode(rp, zbrar0, false, true, m1 + 1)) * P) )(zh0), n0 + 1 < BRANCH_SIZE, m1 + 1 < BRANCH_SIZE, n0 < BRANCH_SIZE, n0 < m1 + 1, btree-sub-clean(r0, n0, zh0), disj(asar, 0, m1 + 1), disj(asar0, 0, m1 + 1), inv(istore), inv(gstore2 \ ginval1), inv(gstore1), inv(istore2), inv(istore0), refsize-cons(istore, as0 ∪ union(asar, 0, m1 + 1)), refsize-cons(istore2, as0 ∪ as), sorted(zbrar0, 0, m1), adr3 ∈ istore2, r0 ∈ zh1, m ∈ istore2, m3 ∈ istore, m ∈ istore, adr3 ∈ alladrs, adr3 ∈ istore, n0 ≤ m1, union(asar, 0, m1 + 1) ∩ dom nodes(istore2) ⊆ alladrs, union(asar, 0, m1 + 1) ∩ dom nodes(istore2) ⊆ as, as ⊆ dom nodes(istore2), as ⊆ alladrs, alladrs ⊆ dom nodes(istore2), as0 ⊆ dom nodes(istore2), istore2 ⊑ istore, istore0 ⊑ istore2, istore0 ⊑ istore, ∀ n. ¬ m1 < n → asar[n] ⊆ dom nodes(istore2), ∀ n. ¬ m1 < n → asar[n] ⊆ dom nodes(istore), ∀ n. ¬ m1 < n → asar[n] ⊆ alladrs, ∀ n. ¬ m1 < n → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, m1 + 1), ∀ height0, zh, r0, r, zbrar, boolvar, m0, adr0, adr1, istore1, istore, pstate, m, istore0, gstore0, nx, ginval, as, as0, gstore, rp, dirty, ri, alladrs, P, adr, ino, ns, err, bv. height0 < height1 +1 → zh[r0] = mkznode(r, zbrar, boolvar, true, m0) ∧ adr0 = adr1 ∧ istore1 = istore ∧ pstate = index(m) ∧ remove-refsize0(istore0) = istore0 ∧ remove-refsize0(gstore0) = gstore0 ∧ elems nx ∩ ginval = ∅ ∧ as ∩ as0 = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ ((btree(rp, r0, adr1, height0, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh) ∧ inv(istore0) ∧ inv(istore) ∧ inv(gstore0) ∧ inv(gstore \ ginval) ∧ refsize-cons(istore, as0 ∪ as) ∧ m ∈ istore ∧ r0 ∈ zh ∧ as0 ⊆ dom nodes(istore) ∧ alladrs ⊆ dom nodes(istore) ∧ as ⊆ alladrs ∧ as ⊆ dom nodes(istore) ∧ istore0 ⊑ istore ∧ ¬ dups(nx) ∧ ¬ adr1 ∈ as ∧ (height0 = 0 → as = ∅) → ⟪btree_commit_rec#(r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → bv) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ r0 ∈ zh ∧ (err = ESUCCESS → ¬ zh[r0].dirty ∧ pstate.index?) ∧ (∃ as1, alladrs1. refsize-cons(istore, as0 ∪ as1) ∧ as1 ∩ dom nodes(istore1) ⊆ as ∧ as1 ∩ as0 = ∅ ∧ alladrs1 ⊆ alladrs ∪ as1 ∧ ((btree(rp, r0, (err = ESUCCESS ⊃ adr1;adr0), height0, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh))) ⊦ ⟪btree_commit_rec#(zbrar0[n0].child, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr, err, zh0, istore, pstate, syn); n := n0; if err = ESUCCESS then zh0[r0].zbranches[n].adr := adr; n := n + 1⟫ ( commit-inv (n, pstate, istore2, istore0, nx, gstore2, gstore1, ginval1, syn, err, rp, r0, adr3, height1 + 1, true, ri, istore, as, as0, alladrs, P, zh0) ∧ ¬ m1 < (zh0[r0].usedsize - n) + n0), # alladrs = 0, zbrar0[n0].child = null, alladrs = ∅, dups(nx), adr3 ∈ as, adr3 ∈ union(asar, 0, m1 + 1)

Rule argument

spec: iter-n, inst: zheap name: Sep-unfold seq: ⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1) subst: [m0, m, n, nP0] → [n0, m1 + 1, 0, λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, m1 + 1, ri), nodes(istore) | union(asar, 0, m1 + 1), asar[n], asar0[n], ok)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n0, m1 + 1, 0, λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, m1 + 1, ri), nodes(istore) | union(asar, 0, m1 + 1), asar[n], asar0[n], ok)